Mortar multiscale ﬁnite element methods for Stokes–Darcy...

Numer. Math. (2014) 127:93–165DOI 10.1007/s00211-013-0583-z

NumerischeMathematik

Mortar multiscale finite element methodsfor Stokes–Darcy flows

Vivette Girault · Danail Vassilev · Ivan Yotov

Received: 25 January 2012 / Revised: 7 March 2013 / Published online: 29 September 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract We investigate mortar multiscale numerical methods for coupled Stokesand Darcy flows with the Beavers–Joseph–Saffman interface condition. The domainis decomposed into a series of subdomains (coarse grid) of either Stokes or Darcytype. The subdomains are discretized by appropriate Stokes or Darcy finite elements.The solution is resolved locally (in each coarse element) on a fine scale, allowing fornon-matching grids across subdomain interfaces. Coarse scale mortar finite elementsare introduced on the interfaces to approximate the normal stress and impose weaklycontinuity of the normal velocity. Stability and a priori error estimates in terms ofthe fine subdomain scale h and the coarse mortar scale H are established for fairlygeneral grid configurations, assuming that the mortar space satisfies a certain inf-supcondition. Several examples of such spaces in two and three dimensions are given.Numerical experiments are presented in confirmation of the theory.

D. Vassilev and I. Yotov were partially supported by the NSF grant DMS 1115856 and the DOE grantDE-FG02-04ER25618.

V. GiraultUPMC-Paris 6 and CNRS, UMR 7598, 75230 Paris Cedex 05, Francee-mail: [email protected]

V. GiraultDepartment of Mathematics, TAMU, College Station, TX 77843, USA

D. VassilevMathematics Research Institute, College of Engineering, Mathematics and Physical Sciences,University of Exeter, Exeter EX4 4QF, UKe-mail: [email protected]

I. Yotov (B)Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USAe-mail: [email protected]

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94 V. Girault et al.

Mathematics Subject Classification 65N12 · 65N15 · 65N30 · 65N55 · 76D07 ·76S05

1 Introduction

Mathematical and numerical modeling of coupled Stokes and Darcy flows has becomea topic of significant interest in recent years. Such coupling occurs in many appli-cations, including surface water-groundwater interaction, flows through vuggy orfractured porous media, industrial filters, fuel cells, and cardiovascular flows. Themost commonly used model is based on the experimentally derived Beavers–Joseph–Saffman interface condition [10,55], a slip with friction condition for the Stokes flowwith a friction coefficient that depends on the permeability of the adjacent porousmedia. Existence and uniqueness of a weak solution has been studied in [24,46].Numerous stable and convergent numerical methods have been developed, see, e.g.,[24,28,29,32,33,44,46,49,54] for methods based on different numerical discretiza-tions suitable for each region, and [4,9,19,45,47,59] for approaches employing unifiedfinite elements. The full Beavers–Joseph condition was considered in [2,20]. A cou-pling of Stokes–Darcy flows with transport was analyzed in [57]. The nonlinear systemof coupled Navier–Stokes and Darcy flows has been studied in [8,26,35].

In this paper we develop multiscale mortar methods for multi-domain non-matchinggrid discretizations of Stokes–Darcy flows in two and three dimensions. Non-matchinggrids provide flexibility in meshing complex geometries with relatively simple locallyconstructed subdomain grids that are suitable for the choice of subdomain discretiza-tion methods. Mortar finite elements play the role of Lagrange multipliers to imposeweakly interface conditions. In [46], a Lagrange multiplier approximating the normalstress was introduced to impose continuity of the normal velocity for discretizationsinvolving mixed finite element methods for Darcy and conforming Stokes elements.With a choice of the Lagrange multiplier space as the normal trace of the Darcy veloc-ity space, the analysis in [46] applied to non-matching grids on the Stokes–Darcyinterface, although this was not explicitly noted. A similar choice was considered insubsequent mortar discretizations for Stokes–Darcy flows [14,29,54]. Mortar methodsfor mixed finite element discretizations for Darcy have been studied in [5,6,52,60].The analysis in these papers allows for the mortar grid to be different from the tracesof the subdomain grids with the assumption that the mortar space satisfies a suitablesolvability condition that limits the number of mortar degrees of freedom. Mortardiscretizations for Stokes have been developed in [11,12]. There, the mortar grid waschosen to be the trace of one of the neighboring subdomain grids, similar to the choicein mortar methods for conforming Galerkin discretizations for second order ellipticproblems [13].

In this work we allow for non-matching grid interfaces of Stokes–Darcy, Stokes–Stokes, and Darcy–Darcy types. We develop multiscale discretizations, where thesubdomains are discretized on a fine scale h and the mortar space is discretized on acoarse scale H . Our method is based on saddle-point formulations in both regions andemploys inf-sup stable mixed finite elements for Darcy and conforming elements forStokes. The mortars approximate different physical variables and are used to imposedifferent matching conditions depending on the type of interface. On Stokes–Stokes

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Mortar multiscale finite element methods for Stokes–Darcy flows 95

interfaces, the mortar functions represent the entire stress vector and impose weak con-tinuity of the entire velocity vector. On Stokes–Darcy and Darcy–Darcy interfaces, themortars approximate the normal stress, which is just the pressure in the Darcy region,and impose weak continuity of the normal velocity. The mortar spaces are assumedto satisfy suitable inf-sup conditions, allowing for very general subdomain and mor-tar grid configurations. We consider approximations of different polynomial degreeson the three types of interfaces and the two types of subdomains. The mortar spacescan be continuous or discontinuous, the latter providing localized mass conservationacross interfaces. Our method is more general than existing Stokes–Stokes mortarmethods [11,12] and Stokes–Darcy mortar methods [14,29,46,54]. On Darcy–Darcyinterfaces, our condition is closely related to the solvability condition considered in[5,6,52,60].

The stability and convergence analysis relies on the construction of a bounded globalinterpolant in the space of weakly continuous velocities that also preserves the velocitydivergence in the usual discrete sense. This is done in two steps, starting from suitablelocal interpolants and correcting them to satisfy the interface matching conditions. Thecorrection step requires the existence of bounded mortar interpolants. This is a verygeneral condition that can be easily satisfied in practice. We present two examples in2-D and one example in 3-D that satisfy this solvability condition. Our error analysisshows that the global velocity and pressure errors are bounded by the fine scale localapproximation error and the coarse scale non-conforming error. Since the polynomialdegrees on subdomains and interfaces may differ, one can choose higher order mortarpolynomials to balance the fine scale and the coarse scale error terms and obtainfine scale asymptotic convergence. The dependence of the stability and convergenceconstants on the subdomain size is explicitly determined. In particular, the stabilityand fine scale convergence constants do not depend on the size of subdomains, whilethe coarse scale non-conforming error constants deteriorate when the subdomain sizegoes to zero. This is to be expected, as the relative effect of the non-conforming errorbecomes more significant in such regime. However, this dependence can be madenegligible by choosing higher order mortar polynomials, as mentioned above.

Our multiscale Stokes–Darcy formulation can be viewed as an extension of themortar multiscale mixed finite element (MMMFE) method for Darcy developed in[6]. The MMMFE method provides an alternative to other multiscale methods in theliterature such as the variational multiscale method [3,41] and the multiscale finiteelement method [22,40]. All three methods utilize a divide and conquer approach:solve relatively small fine scale subdomain problems that are only coupled on thecoarse scale through a reduced number of degrees of freedom. The mortar multiscaleapproach is more flexible as it allows for employment of a posteriori error estimationto adaptively refine the mortar grids where necessary to improve the global accuracy[6]. Following the non-overlapping domain decomposition approach from [37], it canbe shown that the global Stokes–Darcy problem can be reduced to a positive definitecoarse scale interface problem [58]. The latter can be solved using a preconditionedKrylov space solver requiring Stokes or Darcy subdomain solves at each iteration. Analternative more efficient implementation for MMMFE discretizations for Darcy wasdeveloped in [31], where a multiscale flux basis giving the interface flux response foreach coarse scale mortar degree of freedom is precomputed. The multiscale flux basis

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is used to replace the solution of subdomain problems by a simple linear combination.The application of this methodology to the Stokes–Darcy interface problem will bediscussed in a forthcoming paper. We should mention that there have been a number ofpapers in the literature studying domain decomposition methods for the Stokes–Darcyproblem, primarily in the two-subdomain case, see, e.g., [21,25,27,30,39].

1.1 Notation and preliminaries

Let � be a bounded, connected Lipschitz domain of Rn , n = 2, 3, with boundary∂� and exterior unit normal vector n, and let � be a part of ∂� with positive n − 1measure: |�| > 0. We do not assume that � is connected, but if it is not connected, weassume that it has a finite number of connected components. In the case when n = 3,we also assume that � is itself Lipschitz. Let

H10,�(�) = {v ∈ H1(�); v|� = 0}.

Poincaré’s inequality in H10,�(�) reads: There exists a constant P� depending only

on � and � such that

∀v ∈ H10,�(�), ‖v‖L2(�) ≤ P�|v|H1(�). (1.1)

The norms and spaces are made precise later on. The formula (1.1) is a particular caseof a more general result (cf. [15,50]):

Proposition 1.1 Let � be a bounded, connected Lipschitz domain of Rn and let � bea seminorm on H1(�) satisfying:

1) There exists a constant P1 such that

∀v ∈ H1(�), �(v) ≤ P1‖v‖H1(�). (1.2)

2) The condition �(c) = 0 for a constant function c holds if and only if c = 0. Thenthere exists a constant P2 depending only on �, such that

∀v ∈ H1(�), ‖v‖L2(�) ≤ P2(|v|2H1(�)

+ �(v)2) 12 . (1.3)

We recall Korn’s first inequality: There exists a constant C1 depending only on �

and � such that

∀v ∈ H10,�(�)n, |v|H1(�) ≤ C1‖D(v)‖L2(�), (1.4)

where D(v) is the deformation rate tensor, also called the symmetric gradient tensor:

D(v) = 1

2

(∇ v + ∇ vT ).

This is a particular case of the following more general result (see (1.6) in [16]):

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Proposition 1.2 Let � be a bounded, connected Lipschitz domain of Rn and let � bea seminorm on H1(�)n satisfying:

1) There exists a constant C2 such that

∀v ∈ H1(�)n, �(v) ≤ C2‖v‖H1(�). (1.5)

2) The condition �(m) = 0 for a rigid-body motion m holds if and only if m is aconstant vector. Then there exists a constant C3 depending only on �, such that

∀v ∈ H1(�)n, |v|H1(�) ≤ C3(‖D(v)‖2

L2(�)+ �(v)2) 1

2 . (1.6)

In particular, Proposition 1.2 implies that there exists a constant C�, only dependingon � such that (see (1.9) in [16]),

∀v ∈ H1(�)n, |v|H1(�) ≤ C�

⎛

⎜⎝‖D(v)‖2

L2(�)+∣∣∣∣∣∣

∫

�

curl v

∣∣∣∣∣∣

2⎞

⎟⎠

12

, (1.7)

where | · | denotes the Euclidean vector norm.For any non-negative integer m, recall the classical Sobolev space (cf. [1] or [50])

Hm(�) ={v ∈ L2(�); ∂kv ∈ L2(�)∀ |k| ≤ m

},

equipped with the following seminorm and norm (for which it is a Hilbert space)

|v|Hm(�) =⎛

⎝∑

|k|=m

∫

�

|∂kv|2 dx

⎞

⎠

12

, ‖v‖Hm (�) =⎛

⎝∑

0≤|k|≤m

|v|2Hk (�)

⎞

⎠

12

.

This definition is extended to any real number s = m + s′ for an integer m ≥ 0 and0 < s′ < 1 by defining in dimension n the fractional semi-norm and norm

|v|Hs (�) =⎛

⎝∑

|k|=m

∫

�

∫

�

|∂kv(x) − ∂kv( y)|2|x − y|n+2 s′ dx d y

⎞

⎠

12

, ‖v‖Hs (�) =(‖v‖2

Hm (�) + |v|2Hs (�)

) 12.

In particular, we shall frequently use the fractional Sobolev spaces H12 (�) and H

12

00(�)

for a Lipschitz surface � when n = 3 or curve when n = 2 with the followingseminorms and norms:

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|v|H

12 (�)

=⎛

⎝∫

�

∫

�

|v(x) − v( y)|2|x − y|n dxd y

⎞

⎠

12

‖v‖H

12 (�)

=(

‖v‖2L2(�)

+ |v|2H

12 (�)

) 12

,

(1.8)

|v|H

12

00 (�)

=⎛

⎝|v|2H

12 (�)

+∫

�

|v(x)|2d∂�(x)

dx

⎞

⎠

12

‖v‖H

12

00 (�)

=(

‖v‖2L2(�)

+ |v|2H

12

00 (�)

) 12

,

(1.9)

where d∂�(x) denotes the distance from x to ∂�. When � is a subset of ∂� with

positive n − 1 measure, H12

00(�) is the space of traces of all functions of H10,∂�\�(�).

The above norms (1.8) and (1.9) are not equivalent except when � is a closed surface

or curve. The dual space of H12 (�) is denoted by H− 1

2 (�).In addition to these spaces, we shall use the Hilbert space

H(div;�) ={v ∈ L2(�)n; div v ∈ L2(�)

},

equipped with the graph norm

‖v‖H(div;�) =(‖v‖2

L2(�)+ ‖div v‖2

L2(�)

) 12.

The normal trace v · n of a function v of H(div;�) on ∂� belongs to H− 12 (∂�)

(cf. [34]). The same result holds when � is a part of ∂� and is a closed surface. But

if � is not a closed surface, then v · n belongs to the dual of H12

00(�). When v · n = 0on ∂�, we use the space

H0(div;�) = {v ∈ H(div;�); v · n = 0 on ∂�} .

2 Problem statement

2.1 Coupled Stokes and Darcy systems

Let � be partitioned into two non-overlapping regions: the region of the Darcy flow,�d , and the region of the Stokes flow, �s , each one possibly non-connected, but witha finite number of connected components, and with Lipschitz-continuous boundaries∂�d and ∂�s :

� = �d ∪ �s .

Let �d = ∂�d ∩ ∂�, �s = ∂�s ∩ ∂�, �sd = ∂�d ∩ ∂�s . The unit normal vector on�sd exterior to �d , respectively �s , is denoted by nd , respectively ns . In dimension

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three, we assume that �d , �s , and �sd also have Lipschitz-continuous boundaries. Letf d be the gravity force in �d , f s a given body force in �s , let νd > 0, respectivelyνs > 0, be the constant viscosity coefficient of the Darcy, respectively Stokes flow, letK be the rock permeability tensor in �d , let qd be an external source or sink term in�d , and let α > 0 be the slip coefficient in the Beavers–Joseph–Saffman law [10,55],determined by experiment. As far as the data are concerned, we assume on one handthat K is bounded, symmetric and uniformly positive definite in �d : there exist twoconstants, λmin > 0 and λmax > 0 such that

∀x ∈ �d ,∀χ ∈ Rn, λmin|χ |2 ≤ K (x)χ · χ ≤ λmax|χ |2, (2.1)

and we assume on the other hand, that the source qd satisfies the solvability condition

∫

�d

qd dx = 0. (2.2)

The fluid velocity and pressure in �d , respectively �s , are denoted by ud and pd ,respectively by us and ps . The stress tensor of the Stokes flow is denoted by σ (us, ps),

σ (us, ps) = −ps I + 2νs D(us).

In the Darcy region �d , the pair (ud , pd) satisfies

νd K−1ud + ∇ pd = f d in �d , (2.3)

div ud = qd in �d , (2.4)

ud · n = 0 on �d . (2.5)

In the Stokes region �s , the pair (us, ps) satisfies

− div σ (us, ps) ≡ −2νsdiv D(us) + ∇ ps = f s in �s, (2.6)

div us = 0 in �s, (2.7)

us = 0 on �s . (2.8)

These operators suggest to look for (ud , pd) in H(div;�d) × H1(�d) and (us, ps)

in H1(�s)n × L2(�s). The Darcy and Stokes flows are coupled on �sd through the

following interface conditions

us · ns + ud · nd = 0 on �sd , (2.9)

−(σ (us, ps)ns) · ns ≡ ps − 2νs

(D(us)ns

) · ns = pd on �sd , (2.10)

−√

Kl

νsα

(σ (us, ps)ns

) · τ l ≡−√

Kl

α2(

D(us)ns) · τ l =us · τ l , on �sd , 1≤ l ≤n−1,

(2.11)

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where τ l , 1 ≤ l ≤ n − 1 is an orthogonal system of unit tangent vectors on �sd andKl = (Kτ l

)·τ l . Conditions (2.9) and (2.10) incorporate continuity of flux and normalstress, respectively. Condition (2.11) is known as the Beavers–Joseph–Saffman law[10,42,55] describing slip with friction, where

√Kl/α is a friction coefficient.

2.2 First variational formulation

For any functions ϕd defined in �d and ϕs defined in �s , it is convenient to define ϕ

in � by ϕ|�d = ϕd and ϕ|�s = ϕs . With this notation, regarding the data, we assumethat f ∈ L2(�)n , we extend qd by zero in �s , i.e. we set qs = 0 and owing to (2.2),the extended function q belongs to L2

0(�). Before setting problem (2.3)–(2.11) intoan equivalent variational formulation, it is useful to interpret the interface conditions(2.10)–(2.11). First we observe from the regularity of f s that each row of σ (us, ps)

belongs to H(div;�s); hence σ (us, ps)ns belongs to H− 12 (∂�s)

n , and in particular

is well-defined as an element of the dual of H12

00(�sd)n , which is a distribution spaceon �sd . But without further information, it cannot be multiplied directly by the normalor tangent vectors, since the boundary is only Lipschitz-continuous. To bypass thisdifficulty, following [35] we define the function on �sd

g = −pd ns −n−1∑

l=1

νsα√Kl

(us · τ l)τ l , (2.12)

and replace (2.10)–(2.11) by the condition

σ (us, ps)ns = g on �sd . (2.13)

As the traces of pd and of all components of us on �sd belong to H12 (�sd), Sobolev’s

imbeddings [1] imply that g belongs to Lr (�sd)n for any finite r when n = 2 andr = 4 when n = 3. Hence condition (2.13) makes sense. Let us check that it implies(2.10)–(2.11). First, prescribing (2.13) guarantees that σ (us, ps)ns belongs at least toL4(�sd)n and thus can be multiplied by the normal or tangent vectors. Then by virtueof this regularity, (2.12), (2.13) are equivalent to:

((σ (us, ps)ns

) · ns)ns +

n−1∑

l=1

((σ (us, ps)ns

) · τ l)τ l =−pd ns −

n−1∑

l=1

νsα√Kl

(us · τ l)τ l ,

and therefore, by identifying on both sides the components of the normal and tangentvectors (that forms an orthonormal set), we derive (2.10)–(2.11). Hence (2.13) is theinterpretation of (2.10)–(2.11).

Now, let (ud , pd) ∈ H(div;�d) × H1(�d) and (us, ps) ∈ H1(�s)n × L2(�s) be

a solution of (2.3)–(2.11). In order to set (2.3)–(2.11) in variational form, we take thescalar product of (2.3) and (2.6) respectively with any test function vd in H1(�d)n

satisfying vd · n = 0 on �d , and any vs in H1(�s)n satisfying vs = 0 on �s . Then we

apply Green’s formula in �d and �s . This yields

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νd

∫

�d

K−1ud · vd −∫

�d

pddiv vd +∫

�sd

pdvd · nd =∫

�d

f d · vd , (2.14)

2 νs

∫

�s

D(us) : D(vs)−∫

�s

psdiv vs −〈σ (us, ps)ns, vs〉�sd =∫

�s

f s · vs, (2.15)

where 〈·, ·〉�sd denotes the duality pairing between H12

00(�sd)n and its dual space. Thevalidity of (2.14) and (2.15) follows from the above considerations. By summing(2.14) and (2.15), by using the fact that nd = −ns , and by applying (2.13), the termon the interface, say I , reads

I = −〈σ (us, ps)ns, vs〉�sd −∫

�sd

pdvd · ns = −∫

�sd

g · vs −∫

�sd

pdvd · ns .

Then the expression (2.12) for g yields

I =n−1∑

l=1

∫

�sd

νsα√Kl

(us · τ l)(vs · τ l) +∫

�sd

pd [v · n], (2.16)

where the jump [v · n] is defined by

[v · n] = vs · ns + vd · nd .

Finally, we eliminate this jump by enforcing strongly the transmission condition (2.9)on the test function v. In view of the interior terms in (2.14) and (2.15) and whatremains in (2.16), we see that we can reduce the regularity of our functions and workin the space

X = {v ∈ H(div;�); vs ∈ H1(�s)n, v|�s = 0, (v · n)|�d = 0}, (2.17)

which is a Hilbert space equipped with the norm

∀v ∈ X , ‖v‖X = (‖v‖2H(div;�) + |vs |2H1(�s )

) 12 . (2.18)

Note that the restriction of v · n on �sd belongs at least to L4(�sd). Let us denoteW = L2

0(�) with the norm ‖w‖W = ‖w‖L2(�). Then we propose the following

variational formulation: Find (u, p) ∈ X × W such that

∀v ∈ X , νd

∫

�d

K−1ud · vd + 2 νs

∫

�s

D(us) : D(vs) −∫

�

p div v

+n−1∑

l=1

∫

�sd

νsα√Kl

(us · τ l)(vs · τ l) =∫

�

f · v, (2.19)

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∀w ∈ W,

∫

�

w div u =∫

�d

w qd . (2.20)

Lemma 2.1 For any data f in L2(�)n and qd in L20(�d), any solution (u, p) ∈ X×W

of problem (2.19)–(2.20) satisfies pd ∈ H1(�d) and solves (2.3)–(2.11). Conversely,any solution of (2.3)–(2.11), (u, p) ∈ X×W with pd ∈ H1(�d), solves (2.19)–(2.20).

Proof It stems from the above considerations that any solution (ud , pd) ∈ H(div;�d)

×H1(�d) and (us, ps) ∈ H1(�s)n × L2(�s) of (2.3)–(2.11) is such that (u, p)

belongs to X × L2(�) and solves (2.19)–(2.20). Moreover as � is connected, (2.19)only defines p up to an additive constant and this constant can be chosen so that pbelongs to W .

Conversely, let (u, p) ∈ X × W solve (2.19)–(2.20), and denote its restriction to�d and �s as above. By choosing smooth test functions with compact support first in�d and next in �s , we immediately derive that (ud , pd) is a solution of (2.3)–(2.5)and (us, ps) is a solution of (2.6)–(2.8). Furthermore, since both f d and νd K−1ud

belong to L2(�d)n , (2.3) implies that pd ∈ H1(�d).It remains to recover the transmission conditions (2.9)–(2.11). First, (2.9) is a con-

sequence of the definition (2.17) of X . Next, we recover (2.14) and (2.15) by takingthe scalar product of (2.3) and (2.6) with a function v ∈ X that is smooth in �d andin �s , and by applying Green’s formula in both regions. By comparing with (2.19),this gives

∫

�sd

pdvd · nd − 〈σ (us, ps)ns, vs〉�sd =n−1∑

l=1

∫

�sd

νsα√Kl

(us · τ l)(vs · τ l).

By taking into account the orientation of the normal, the regularity of v, and thedefinition (2.12) of g, this is equivalent to:

〈σ (us, ps)ns, vs〉�sd = −∫

�sd

pdvs · ns −n−1∑

l=1

∫

�sd

νsα√Kl

(us · τ l)(vs · τ l) =∫

�sd

g · vs .

As the trace space of vs on �sd is large enough, this implies (2.13).

2.3 Existence and uniqueness of the solution

For any functions ud , vd in L2(�d)n and us , vs in H1(�s)n , we define the bilinear

form

a(u, v) = νd

∫

�d

K−1ud · vd + 2 νs

∫

�s

D(us) : D(vs) +n−1∑

l=1

∫

�sd

νsα√Kl

(us · τ l )(vs · τ l ),

(2.21)

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and for any functions vd ∈ H(div;�d), vs ∈ H(div;�s) and w ∈ L2(�), we definethe bilinear form

b(v, w) = −∫

�d

w div vd −∫

�s

w div vs . (2.22)

Note that a(·, ·) is continuous on X × X and b(·, ·) is continuous on X × L2(�):

∀(u, v) ∈ X × X , |a(u, v)| ≤ νd

λmin‖ud‖L2(�d )‖vd‖L2(�d )

+2 νs‖∇ us‖L2(�s )‖∇ vs‖L2(�s )

+n−1∑

l=1

νsα√λmin

‖us · τ l‖L2(�sd )‖vs · τ l‖L2(�sd ),

∀(v, w) ∈ X × L2(�), |b(v, w)| ≤ ‖v‖X‖w‖L2(�). (2.23)

Then (2.19)–(2.20) has the familiar form : Find (u, p) ∈ X × W such that

∀v ∈ X , a(u, v) + b(v, p) =∫

�

f · v, (2.24)

∀w ∈ W, b(u, w) = −∫

�d

w qd . (2.25)

Next, we set

X0 = {v ∈ X; div v = 0}, (2.26)

and more generally, for a given function g ∈ W , we define the affine variety

Xg = {v ∈ X; div v = g}. (2.27)

Then we consider the reduced problem : Find u ∈ Xq such that

∀v ∈ X0, a(u, v) =∫

�

f · v, (2.28)

recall that q is qd extended by zero on �s . It is well known [18,34] that showing equiv-alence between problems (2.28) and (2.24)–(2.25) amounts to proving the followinginf-sup condition.

Lemma 2.2 There exists a constant β > 0 such that

∀w ∈ W, supv∈X

b(v, w)

‖v‖X

≥ β‖w‖W . (2.29)

123


Proof Let w ∈ W . The inf-sup condition between H10 (�)n and L2

0(�) implies thatthere exists a function v ∈ H1

0 (�)n such that

div v = w in � and |v|H1(�) ≤ 1

κ‖w‖L2(�),

where κ depends only on �; see for example [34] or [18]. Then v belongs to X and a

straightforward computation yields (2.29) with β ≥ κ/(P2

∂� + 2) 1

2 , where P∂� is thePoincaré constant in (1.1).

Lemma 2.2 implies that (2.28) and (2.24)–(2.25) are equivalent in the following sense.

Proposition 2.1 Let ( f , qd) be given in L2(�)n × L20(�d). Let (u, p) ∈ X × W be a

solution of (2.24)–(2.25). Then u solves (2.28). Conversely, let u ∈ Xqd be a solutionof (2.28). Then there exists a unique p in W such that (u, p) satisfies (2.24)–(2.25).

Now, let us prove that (2.28), and hence (2.24)–(2.25), is well-posed. This relies onthe ellipticity of a(·, ·) on X0. If �s is connected and |�s | > 0, a partial ellipticityresult for a(·, ·) follows directly from Korn’s inequality (1.4):

∀v ∈ X , a(v, v) ≥ 2νs

C21

|vs |2H1(�s )+ νd

λmax‖vd‖2

L2(�d ), (2.30)

with the constant C1 of (1.4). If �s is connected and |�s | = 0, then �sd = ∂�s up toa set of zero measure, and proving the analogue of (2.30) makes use of (1.7) and thetangential components on �sd . Indeed, we have formally

a.e. on ∂�s,∣∣(vs × ns)(x)

∣∣ ≤n−1∑

l=1

|(vs · τ l)(x)|, (2.31)

and therefore

∫

�s

curl vs = −∫

�sd

vs × ns ⇒∣∣∣∫

�s

curl vs

∣∣∣ ≤∫

�sd

( n−1∑

l=1

|vs · τ l |).

Hence (1.7) and a straightforward manipulation yield for all vs in H1(�s)n :

2νs‖D(vs)‖2L2(�s )

+n−1∑

l=1

νsα√Kl

‖vs · τ l‖2L2(�sd )

≥ νs

C2�

min

(2,

α√λmax|�sd |

)|vs |2H1(�s )

. (2.32)

123


As a consequence (2.30) is replaced by

∀v ∈ X , a(v, v) ≥ νd

λmax‖vd‖2

L2(�d )+ νs

C2�

min

(2,

α√λmax|�sd |

)|vs |2H1(�s )

.

(2.33)

Finally, if �s is not connected, the analogue of (2.30) holds on all connected compo-nents of �s that are adjacent to �s and the analogue of (2.33) holds on all connectedcomponents of �s that are not adjacent to �s .

It remains to establish that the mapping:

v �→ |v|X0= (|vs |2H1(�s)

+ ‖vd‖2L2(�d )

) 12 (2.34)

is a norm on X0 equivalent to ‖v‖X . This is the object of the next lemma.

Lemma 2.3 There exists a constant C4 such that

∀v ∈ X0, ‖vs‖L2(�s )≤ C4

(|vs |2H1(�s)+ ‖vd‖2

L2(�d )

) 12 . (2.35)

Proof Let us assume that �s is connected; the case when �s is not connected is treatedas above. If |�s | > 0, (2.35) follows from Poincaré’s inequality (1.1) applied in �s

and does not require the norm of vd in the right-hand side. When |�s | = 0, the proofof (2.35) is a variant of the proof of Peetre–Tartar’s Lemma [51]. Let us recall itsargument. Assume that (2.35) is not true. Then there exists a sequence (vm) in X0such that

limm→∞ ‖vm

d ‖L2(�d ) = limm→∞ |vm

s |H1(�s )= 0 and ∀m, ‖vm

s ‖L2(�s )= 1.

As X0 is reflexive, this implies that there exists a function v ∈ X0 such that

limm→∞ vm = v weakly in X .

Moreover vs = c, a constant vector, and vd = 0. Then the fact that v belongs to Ximplies that c · ns = 0 on �sd , and since �sd coincides a.e. with ∂�s , this implies thatc = 0. Thus

limm→∞ vm

s = 0 weakly in H1(�s)n,

hence

limm→∞ vm

s = 0 strongly in L2(�s)n .

This contradicts the fact that for all m ‖vms ‖L2(�s )

= 1.

Therefore (2.35) combined with either (2.30) or (2.33) yields the ellipticity of a(·, ·).

123


Fig. 1 Domain decompositionwith multiple connectedcomponents

Lemma 2.4 There exists a constant C5 > 0 such that

∀v ∈ X0, a(v, v) ≥ C5‖v‖2X. (2.36)

With the continuity of a(·, ·) and b(·, ·), the ellipticity of a(·, ·) on X0, and the inf-sup condition (2.29), the Babuška–Brezzi’s theory [7,17] implies immediately that(2.24)–(2.25) is well-posed.

Theorem 2.1 Problem (2.24)–(2.25) has a unique solution (u, p) ∈ X ×W and thereexists a constant C that depends only on �d , �s , λmin, λmax, α, νd , and νs , such that

‖u‖X + ‖p‖L2(�) ≤ C(‖ f ‖L2(�) + ‖qd‖L2(�d )

). (2.37)

In turn, the well-posedness of problem (2.3)–(2.11) stems from Lemma 2.1.

2.4 Domain decomposition of the Darcy and Stokes regions

Let �s , respectively �d , be decomposed into Ms , respectively Md , non-overlapping,open Lipschitz subdomains:

�s = ∪Msi=1�s,i , �d = ∪Md

i=1�d,i .

Set M = Md + Ms ; according to convenience we can also number the subdomainswith a single index i , 1 ≤ i ≤ M , the Darcy subdomains running from Ms +1 to M . Anexample of a domain decomposition with multiple connected components is depictedin Fig. 1. Let ni denote the outward unit normal vector on ∂�i . For 1 ≤ i ≤ M , letthe boundary interfaces be denoted by �i , with possibly zero measure:

�i = ∂�i ∩ ∂�,

and for 1 ≤ i < j ≤ M , let the interfaces between subdomains be denoted by �i j ,again with possibly zero measure:

�i j = ∂�i ∩ ∂� j .

123


In addition to �sd , let �dd , respectively �ss , denote the set of interfaces between subdo-mains of �d , respectively �s . Then, assuming that the solution (u, p) of (2.3)–(2.11)is slightly smoother, we can obtain an equivalent formulation by writing individually(2.3)–(2.11) in each subdomain �i , 1 ≤ i ≤ M , and complementing these systemswith the following interface conditions

[uud · n] = 0, [pd ] = 0 on �dd , (2.38)

[us] = 0, [σ (us, ps)n] = 0 on �ss, (2.39)

where the jumps on an interface �i j , 1 ≤ i < j ≤ M , are defined as

[v · n] = vi · ni + v j · n j , [σ n] = σ i ni + σ j n j , [v] = (vi − v j )|�i j ,

using the notation vi = v|�i . The smoothness requirement on the solution is meantto ensure that the jumps [ud · n], respectively [σ (us, ps)n], are well-defined on eachinterface of �dd , respectively �ss .

Finally, let us prescribe weakly the interface conditions (2.38), (2.39), and (2.9)by means of Lagrange multipliers, usually called mortars. For this, it is convenient toattribute a unit normal vector ni j to each interface �i j of positive measure, directedfrom �i to � j (recall that i < j). The basic velocity spaces are:

Xd = {v ∈ L2(�d)n; vd,i := v|�d,i ∈ H(div;�d,i ), 1 ≤ i ≤ Md ,

v · ni j ∈ H− 12 (�i j ), �i j ∈ �dd ∪ �sd , v · n = 0 on �d}, (2.40)

Xs = {v ∈ L2(�s)n; vs,i := v|�s,i ∈ H1(�s,i )

n, 1 ≤ i ≤ Ms, v = 0 on �s},

and the mortar spaces are:

∀�i j ∈ �ss, �i j = H− 12 (�i j )

n, ∀�i j ∈ �sd ∪ �dd , �i j = H12 (�i j ).

(2.41)

Then we replace X (see (2.17)) by

X = {v ∈ L2(�)n; vd := v|�d ∈ Xd , vs := v|�s ∈ Xs}, (2.42)

we keep W = L20(�) for the pressure, and we define the mortar spaces

�s = {λ ∈ (D′(�ss))n; λ|�i j ∈ H− 1

2 (�i j )n for all �i j ∈ �ss},

�sd = {λ ∈ L2(�sd); λ|�i j ∈ H12 (�i j ) for all �i j ∈ �sd}, (2.43)

�d = {λ ∈ L2(�dd); λ|�i j ∈ H12 (�i j ) for all �i j ∈ �dd}.

123


We equip these spaces with broken norms:

|||v|||Xd=⎛

⎝Md∑

i=1

‖v‖2H(div;�d,i )

⎞

⎠

12

, |||v|||Xs=( Ms∑

i=1

‖v‖2H1(�s,i )

) 12

,

|||v|||X =(|||v|||2Xd

+ |||v|||2Xs

) 12, |||λ|||�s

=⎛

⎝∑

�i j ∈�ss

‖λ‖2

H− 12 (�i j )

⎞

⎠

12

,

|||λ|||�sd=⎛

⎝∑

�i j ∈�sd

‖λ‖2

H12 (�i j )

⎞

⎠

12

, |||λ|||�d=⎛

⎝∑

�i j ∈�dd

‖λ‖2

H12 (�i j )

⎞

⎠

12

.

Note that in most geometrical situations, Xd (and hence X ) is not complete for theabove norm, but none of the subsequent proofs require its completeness.

The matching condition between subdomains is weakly enforced through the fol-lowing bilinear forms:

∀v ∈ Xs,∀μ ∈ �s, bs(v,μ) =∑

�i j ∈�ss

〈[v],μ〉�i j ,

∀v ∈ Xd ,∀μ ∈ �d , bd(v, μ) =∑

�i j ∈�dd

〈[v · n], μ〉�i j , (2.44)

∀v ∈ X,∀μ ∈ �sd , bsd(v, μ) =∑

�i j ∈�sd

〈[v · n], μ〉�i j .

For the velocity and pressure in the Darcy and Stokes regions, we use the followingbilinear forms:

∀(u, v) ∈ Xs ×Xs, as,i (u, v) = 2 νs

∫

�s,i

D(us,i ) : D(vs,i )

+n−1∑

l=1

∫

∂�s,i ∩�sd

νsα√Kl

(us ·τ l)(vs ·τ l), 1≤ i ≤ Ms,

∀(u, v) ∈ Xd × Xd , ad,i (u, v) = νd

∫

�d,i

K−1ud,i · vd,i , 1 ≤ i ≤ Md , (2.45)

∀v ∈ X,∀w ∈ L2(�), bi (v, w) = −∫

�i

wdiv vi , 1 ≤ i ≤ M.

Then we set

∀(u, v) ∈ X × X, a(u, v) =Ms∑

i=1

as,i (u, v) +Md∑

i=1

ad,i (u, v),

123


∀(v, w) ∈ X × L2(�), b(v, w) =M∑

i=1

bi (v, w).

The second variational formulation reads: Find (u, p, λsd , λd ,λs) ∈ X × W ×�sd ×�d × �s such that

∀v ∈ X, a(u, v) + b(v, p) + bsd(v, λsd) + bd(v, λd) + bs(v,λs) =∫

�

f · v,

∀w ∈ W, b(u, w) = −∫

�d

w qd , (2.46)

∀μ ∈ �sd , bsd(u, μ) = 0, ∀μ ∈ �d , bd(u, μ) = 0, ∀μ ∈ �s, bs(u,μ) = 0.

It remains to prove that (2.46) is equivalent to (2.3)–(2.11) when the solution issufficiently smooth. Since we know from Theorem 2.1 that (2.3)–(2.11) has a uniquesolution, equivalence will also establish that (2.46) is uniquely solvable.

Theorem 2.2 Assume that the solution (u, p) ∈ X × W of (2.3)–(2.11) with pd ∈H1(�d) satisfies

∀�i j ∈ �dd ∪ �sd , (ud · nd)|�i j ∈ H− 12 (�i j ), ∀�i j ∈ �ss,

(σ (us, ps)ns)|�i j ∈ H− 12 (�i j )

n .

Then (2.46) is equivalent to (2.3)–(2.11).

Proof The argument is similar to that used in proving Lemma 2.1. Let (u, p) be asolution of (2.3)–(2.11) satisfying the above regularity. Take the scalar product in each�i of (2.3) and (2.6) with a test function v in X , apply Green’s formula and add thecorresponding equations. In view of (2.16) and the regularity of (u, p), this gives:

a(u, v) + b(v, p) −∑

�i j ∈�ss

〈σ (us , ps)ni j , [v]〉�i j +∑

�i j ∈�dd∪�sd

〈pd , [v · n]〉�i j =∫

�

f · v.

(2.47)

We recover the first equation in (2.46) by defining

∀�i j ∈ �ss, λs |�i j = −σ (us, ps)|�i j ni j ,

∀�i j ∈ �dd , λd |�i j = pd |�i j , ∀�i j ∈ �sd , λsd |�i j = pd |�i j ,(2.48)

and the remaining equations follow from the regularity of (u, p).Conversely, let (u, p, λsd , λd ,λs) in X × W × �sd × �d × �s be a solution of

(2.46). By choosing smooth test functions with compact support in each �i we recoverthe interior equations (2.3), (2.4), (2.6), (2.7) in each subdomain. On one hand, weeasily derive from the last equation of (2.46) that u has no jump through the interfaces

123


of �ss . Hence u ∈ H1(�s)n . On the other hand, we pick an index i with 1 ≤ i ≤ Ms

and an interface �i j in �ss , we take a function v in X , smooth in �i , zero outside �i ,and zero on ∂�i\�i j . By taking the scalar product of (2.6) in �i with v, applyingGreen’s formula, comparing with (2.46), and using the same process in � j , we find

λs |�i j = −σ (us,i , ps,i )|�i j ni j = −σ (us, j , ps, j )|�i j ni j .

As λs is single-valued, this implies that σ (us, ps)|�i j ni j has no jump through �i j .This is true for all interfaces in �ss . Therefore (2.6) is satisfied in �s . By applyinga similar process to the interfaces of �dd , we derive first that (ud , pd) belongs toH(div;�d)× H1(�d) and next that (2.3) is satisfied in �d . Finally, the third equationin (2.46) implies that u belongs to H(div;�), therefore u is in X and the pair (u, p)

solves (2.19)–(2.20); by virtue of Lemma 2.1, it also solves (2.3)–(2.11).

3 Discretization

3.1 Meshes and discrete spaces

In view of discretization, we assume from now on that � and all its subdomains �i ,1 ≤ i ≤ M , have polygonal or polyhedral boundaries. Since none of the subdomainsoverlap, they form a mesh, Td of �d and Ts of �s , and the union of these meshesconstitutes a mesh T� of �. Furthermore, we suppose that this mesh satisfies thefollowing assumptions:

Hypothesis 3.1 1. T� is conforming, i.e. it has no hanging nodes.2. The subdomains of T� can take at most L different geometrical shapes, where L

is a fixed integer independent of M.3. T� is shape-regular in the sense that there exists a real number σ , independent of

M such that

∀i, 1 ≤ i ≤ M,diam(�i )

diam(Bi )≤ σ, (3.1)

where diam(�i ) is the diameter of �i and diam(Bi ) is the diameter of the largestball contained in �i . Without loss of generality, we can assume that diam(�i ) ≤ 1.

As each subdomain �i is polygonal or polyhedral, it can be entirely triangulated.Let h > 0 denote a discretization parameter, and for each h, let T h

i be a regular familyof partitions of �i made of triangles or tetrahedra T in the Stokes region and triangles,tetrahedra, parallelograms, or parallelepipeds in the Darcy region, with no matchingrequirement at the subdomains interfaces. Thus the meshes are independent and theparameter h < 1 is allowed to vary with i , but to reduce the notation, unless necessary,we do not indicate its dependence on i . By regular, we mean that there exists a realnumber σ0, independent of i and h such that

123


∀i, 1 ≤ i ≤ M, ∀T ∈ T hi ,

hT

ρT≤ σ0, (3.2)

where hT is the diameter of T and ρT is the diameter of the largest ball containedin T .

In addition we assume that each element of T hi has at least one vertex in �i . For

the interfaces, let H > 0 be another discretization parameter and for each H and eachi < j , let T H

i j denote a regular family of partitions of �i j into segments, triangles orparallelograms of diameter bounded by H . These partitions may not match the tracesof the subdomain grids.

On these meshes, we define the following finite element spaces. In the Stokesregion, for each �s,i , let (Xh

s,i , W hs,i ) ⊂ H1(�s,i )

n × L2(�s,i ) be a pair of finiteelement spaces satisfying a local uniform inf-sup condition for the divergence. Moreprecisely, setting Xh

0,s,i = Xhs,i ∩ H1

0 (�s,i )n and W h

0,s,i = W hs,i ∩ L2

0(�s,i ), we assumethat there exists a constant β�

s > 0, independent of h and the diameter of �s,i , suchthat

∀i, 1 ≤ i ≤ Ms, infwh∈W h

0,s,i

supvh∈Xh

0,s,i

∫�s,i

whdiv vh

|vh |H1(�s,i )‖wh‖L2(�s,i )

≥ β�s . (3.3)

In addition, since Xh0,s,i ⊂ H1

0 (�s,i )n , it satisfies a Korn inequality: There exists a

constant α� > 0, independent of h and the diameter of �s,i , such that

∀i, 1 ≤ i ≤ Ms, ∀vh ∈ Xh0,s,i , ‖D(vh)‖L2(�s,i )

≥ α�|vh |H1(�s,i ). (3.4)

There are well-known examples of pairs satisfying (3.3) (cf. [34]), such as the mini-element, the Bernardi–Raugel element, or the Taylor-Hood element. Similarly, in theDarcy region, for each �d,i , let (Xh

d,i , W hd,i ) ⊂ H(div;�d,i ) × L2(�d,i ) be a pair of

mixed finite element spaces satisfying a uniform inf-sup condition for the divergence.More precisely, setting Xh

0,d,i = Xhd,i ∩ H0(div;�d,i ) and W h

0,d,i = W hd,i ∩ L2

0(�d,i ),we assume that there exists a constant β�

d > 0 independent of h and the diameter of�d,i , such that

∀i, 1 ≤ i ≤ Md , infwh∈W h

0,d,i

supvh∈Xh

0,d,i

∫�d,i

whdiv vh

‖vh‖H(div;�d,i )‖wh‖L2(�d,i )

≥ β�d . (3.5)

Furthermore, we assume that

∀i, 1 ≤ i ≤ Md , ∀vh ∈ Xhd,i , div vh ∈ W h

d,i . (3.6)

Again, there are well-known examples of pairs satisfying (3.5) and (3.6) (cf. [18] or[34]), such as the Raviart–Thomas elements, the Brezzi–Douglas–Marini elements,the Brezzi–Douglas–Fortin–Marini elements, the Brezzi–Douglas–Durán–Fortin ele-ments, or the Chen–Douglas elements. Then we discretize straightforwardly Xd andXs by

123


Xhd = {v ∈ L2(�d)n; v|�d,i ∈ Xh

d,i , 1 ≤ i ≤ Md , v · n = 0 on �d},Xh

s = {v ∈ L2(�s)n; v|�s,i ∈ Xh

s,i , 1 ≤ i ≤ Ms, v = 0 on �s},

and we set

W hd = {w ∈ L2(�d); w|�d,i ∈ W h

d,i }, W hs = {w ∈ L2(�s); w|�s,i ∈ W h

s,i },Xh = {v ∈ L2(�)n; v|�d ∈ Xh

d , v|�s ∈ Xhs },

W h = {w ∈ L20(�); w|�d ∈ W h

d , w|�s ∈ W hs }.

The finite elements regularity implies that Xhd ⊂ Xd , Xh

s ⊂ Xs and Xh ⊂ X . Ofcourse, W h ⊂ W .

In the mortar region, we take finite element spaces �Hs , �H

d , and �Hsd . These spaces

consist of continuous or discontinuous piecewise polynomials. We will allow for vary-ing polynomial degrees on different types of interfaces. Although the mortar meshesand the subdomain meshes so far are unrelated, we need compatibility conditionsbetween �H

s , �Hsd and �H

d on one hand, and Xhd and Xh

s on the other hand. Thefollowing set of conditions is fairly crude and will be sharpened further on.

1. For all �i j ∈ �ss ∪ �sd , i < j , and for all v ∈ X , there exists vh ∈ Xhs,i , vh = 0

on ∂�s,i\�i j satisfying

∫

�i j

vh · ni j =∫

�i j

v · ni j . (3.7)

2. For all �i j ∈ �ss , i < j , and for all v ∈ X , there exists vh ∈ Xhs, j , vh = 0 on

∂�s, j\�i j satisfying

∀μH ∈ �Hs ,

∫

�i j

μH · vh =∫

�i j

μH · v. (3.8)

3. For all �i j ∈ �dd ∪ �sd , i < j , and for all v ∈ X , there exists vh ∈ Xhd, j ,

vh · n j = 0 on ∂�d, j\�i j satisfying

∀μH ∈ �Hd , ∀μH ∈ �H

sd ,

∫

�i j

μH vh · ni j =∫

�i j

μH v · ni j . (3.9)

Condition (3.7) is very easy to satisfy in practice and it trivially holds true for allexamples of Stokes spaces considered in this paper. Conditions (3.8) and (3.9) statethat the mortar space is controlled by the traces of the subdomain velocity spaces.Both conditions are easier to satisfy for coarser mortar grids. Condition (3.8) is moregeneral than previously considered in the literature for mortar discretizations of theStokes equations [11,12]. Examples for which (3.8) holds are given in the Appendix:

123


Sect. 7.1 in 2-D and Sect. 7.2 in 3-D. The condition (3.9) on �dd is closely relatedto the mortar condition for Darcy flow in [5,6,52,60] and on �sd , it is more generalthan existing mortar discretizations for Stokes–Darcy flows on [14,29,46,54]. It isdiscussed in more detail in Sect. 4.4.

Lemma 3.1 Under assumptions (3.8) and (3.9), the only solution(λH

sd , λHd ,λH

s

)in

�Hsd × �H

d × �Hs to the system

∀vh ∈ Xh, bs(vh,λH

s ) + bd(vh, λHd ) + bsd(vh, λH

sd) = 0 (3.10)

is the zero solution.

Proof Consider any �i j ∈ �ss with i < j ; the proof for the other interfaces being thesame. Take an arbitrary v in H1

0 (�)n and vh associated with v by (3.8), extended byzero outside �s, j . Then on one hand,

∫

�i j

λHs · v =

∫

�i j

λHs · vh = bs(v

h,λHs ),

and on the other hand,

bd(vh, λHd ) = bsd(vh, λH

sd) = 0.

Therefore

∀v ∈ H10 (�)n,

∫

�i j

λHs · v = 0,

thus implying that λHs = 0.

Finally, we define

V hd ={v∈ Xh

d ; ∀μ ∈ �Hd , bd(v, μ)=0}, V h

s ={v ∈ Xhs ; ∀μ∈�H

s , bs(v,μ)=0},V h ={v∈ Xh; v|�d ∈V h

d , v|�s ∈ V hs ,∀μ ∈ �H

sd , bsd(v, μ)=0}, (3.11)

Zh = {v ∈ V h; ∀w ∈ W h, b(v, w) = 0}.

3.2 Variational formulations and uniform stability of the discrete problem

The discrete version of the second variational formulation (2.46) is: Find (uh, ph, λHsd ,

λHd ,λH

s ) ∈ Xh × W h × �Hsd × �H

d × �Hs such that

∀vh ∈ Xh , a(uh , vh)+b(vh , ph)+bsd (vh , λHsd ) + bd (vh , λH

d ) + bs(vh, λH

s ) =∫

�

f · vh ,

123


∀wh ∈ W h, b(uh , wh) = −∫

�d

wh qd , (3.12)

∀μH ∈ �Hsd , bsd (uh , μH ) = 0, ∀μH ∈ �H

d , bd (uh , μH ) = 0,

∀μH ∈ �Hs , bs(uh ,μH ) = 0.

The last three equations of (3.12) state that uh ∈ V h . Therefore, we can extract from(3.12) the following reduced formulation: Find uh ∈ V h , ph ∈ W h such that

∀vh ∈ V h, a(uh, vh) + b(vh, ph) =∫

�

f · vh,

∀wh ∈ W h, b(uh, wh) = −∫

�d

wh qd .

(3.13)

Lemma 3.2 Problems (3.12) and (3.13) are equivalent.

Proof Clearly, (3.12) implies (3.13). Conversely, if the pair (uh, ph) solves (3.13),existence of λH

sd , λHd ,λH

s such that all these variables satisfy (3.12) is an easy conse-quence of Lemma 3.1 and an algebraic argument.

In view of this equivalence, it suffices to analyze problem (3.13). From the Babuška–Brezzi’s theory, uniform stability of the solution of (3.13) stems from an ellipticityproperty of the bilinear form a in Zh and an inf-sup condition of the bilinear form b.Let us prove an ellipticity property of the bilinear form a, valid when n = 2, 3. Forthis, we make the following assumptions on the mortar spaces:

Hypothesis 3.2 1. On each �i j ∈ �dd ∪ �sd , �Hd |�i j and �H

sd |�i j contain at leastP0.

2. On each �i j ∈ �ss , on each hyperplane F ⊂ �i j , �Hs |F contains at least Pn

0 .3. On each �i j ∈ �ss , �H

s |�i j contains at least Pn1 .

The second assumption guarantees that ni j ∈ �Hs |�i j ; it follows from the third assump-

tion when �i j is flat. The third assumption is solely used for deriving a discrete Korninequality; it can be relaxed, as we shall see in Sect. 7.2. The first two assumptionsimply that all functions vh in V h satisfy

M∑

i=1

∫

�i

div vh =M∑

i=1

∫

∂�i

vh · ni =∑

i< j

∫

�i j

[vh · n] = 0.

Therefore, the zero mean-value restriction on the functions of W h can be relaxed.Thus the condition vh ∈ Zh and (3.6) imply in particular that div vh

d = 0 in �d,i ,1 ≤ i ≤ Md . Hence

∀vh ∈ Zh, |||vhd |||Xd

= ‖vhd‖L2(�d ). (3.14)

First, we treat the simpler case when |�s | > 0 and �s is connected.

123


Lemma 3.3 Let |�s | > 0 and �s be connected. Then under Hypothesis 3.2, we have

∀vh ∈ Zh, a(vh, vh) ≥ νd

λmax|||vh

d |||2Xd+ 2

νs

C2 |||vhs |||2Xs

, (3.15)

where the constant C only depends on the shape regularity of Ts .

Proof Since vhs ∈ V h

s and Pn1 ∈ �H

ss |�i j for each �i j ∈ �ss , then P1[vhs ] = 0, where

P1 is the orthogonal projection on Pn1 for the L2 norm on each �i j . Thus, as �s is

connected and |�s | > 0, inequality (1.12) in [16] gives

∀vhs ∈ V h

s ,

Ms∑

i=1

|vhs |2H1(�s,i )

≤ C2Ms∑

i=1

‖D(vhs )‖2

L2(�s,i ), (3.16)

where the constant C only depends on the shape regularity of Ts . Hence we have theanalogue of (2.30):


λmax|||vh

d |||2Xd+ 2

νs

C2

Ms∑

i=1

|vhs |2H1(�s,i )

. (3.17)

Finally the above argument permits to apply formula (1.3) in [15] in order to recover thefull norm of Xs in the right-hand side of (3.17). In fact, it is enough that Pn

0 ∈ �Hss |�i j

for each �i j ∈ �ss .

Now we turn to the case when �s is connected and |�s | = 0, consequently �sd =∂�s , up to a set of zero measure.

Lemma 3.4 Let |�s | = 0 and �s be connected, i.e. �sd = ∂�s . Then under Hypoth-esis 3.2, we have


λmax|||vh

d |||2Xd+ νs

C2 min

(2,

α√λmax|�sd |

)|||vh

s |||2Xs,

(3.18)

where the constant C only depends on the shape regularity of Ts .

Proof All constants in this proof only depend on the shape regularity of Ts . Since(3.14) holds, it suffices to derive a lower bound for as(v

hs , vh

s ). To begin with, asvh

s ∈ V hs , we apply Theorem 4.2 if n = 2 or 5.2 if n = 3 in [16]:

∀vhs ∈ V h

s ,

Ms∑

i=1

|vhs |2H1(�s,i )

≤ C2

( Ms∑

i=1

‖D(vhs )‖2

L2(�s,i )+ (�(vh

s ))2)

,

123


where the functional � is a suitable seminorm. Let us choose

∀v ∈ Xs, �(v) =

∣∣∣∣∣∣∣

∫

�sd

v × ns

∣∣∣∣∣∣∣. (3.19)

Clearly, � is a seminorm on Xs . Next, considering that

∀v ∈ H1(�s)n, �(v) =

∣∣∣∣∣∣∣

∫

�s

curl v

∣∣∣∣∣∣∣,

it is easy to check that if m is a rigid body motion, then �(m) = 0 if and only if mis a constant vector. Finally, observing that � behaves exactly like the functional �2of Example 2.4 in [16], we see that � satisfies all assumptions of Theorem 4.2 or 5.3in [16]. Thus

∀vhs ∈ V h

s ,

Ms∑

i=1

|vhs |2H1(�s,i )

≤ C2

⎛

⎜⎝

Ms∑

i=1

‖D(vhs )‖2

L2(�s,i )+

∣∣∣∣∣∣∣

∫

�sd

vhs × ns

∣∣∣∣∣∣∣

2⎞

⎟⎠ .

(3.20)

In order to recover the full norm of Xs in the left-hand side of (3.20), we apply Theorem5.1 in [15] with the functional

�(v) =n−1∑

l=1

∫

�sd

|vs · τ l |.

Again, � is a seminorm on Xs and since �sd is a closed curve or surface, the condition�(c) = 0 for a constant vector c implies that c = 0. The remaining assumptions ofthis theorem easily follow by observing that � has the same behavior as the functional�1 of Example 4.2 in [15]. This yields

∀vhs ∈V h

s , ‖vhs ‖2

L2(�s )≤C2

⎛

⎜⎝

Ms∑

i=1

|vhs |2H1(�s,i )

+⎛

⎜⎝

n−1∑

l=1

∫

�sd

|vhs · τ l |

⎞

⎟⎠

2⎞

⎟⎠ . (3.21)

By combining (3.20) and (3.21), we obtain

∀vhs ∈V h

s , |||vhs |||2Xs

≤ C2

⎛

⎜⎜⎝

Ms∑

i=1

‖D(vhs )‖2

L2(�s,i )+

∣∣∣∣∣∣∣

∫

�sd

vhs × ns

∣∣∣∣∣∣∣

2

+⎛

⎜⎝

n−1∑

l=1

∫

�sd

|vhs · τ l |

⎞

⎟⎠

2⎞

⎟⎟⎠.

(3.22)

Then (3.18) follows from (3.22) by arguing as in deriving (2.31).

123


The case when �s is not connected follows from Lemmas 3.3 or 3.4 applied to eachconnected component of �s according that it is or is not adjacent to �s .

To control the bilinear form b in �s , we make the following assumption: There existsa linear approximation operator �h

s : H10 (�)n �→ V h

s satisfying for all v ∈ H10 (�)n :

–

∀i, 1 ≤ i ≤ Ms,

∫

�s,i

div(�h

s (v) − v) = 0. (3.23)

– For any �i j in �sd ,

∫

�i j

(�h

s (v) − v) · ni j = 0. (3.24)

– There exists a constant C independent of v, h, H , and the diameter of �s,i , 1 ≤i ≤ Ms , such that

|||�hs (v)|||Xs

≤ C |v|H1(�). (3.25)

A general construction strategy discussed in Sect. 4.1 gives an operator �hs that satisfies

(3.23) and (3.24). The stability bound (3.25) is shown to hold for the specific examplespresented in Sects. 7.1 and 7.2.

Lemma 3.5 Assume that an operator �hs satisfying (3.23)–(3.25) exists; then there

exists a linear operator �hs : H1

0 (�)n �→ V hs such that for all v ∈ H1

0 (�)n,

∀wh ∈ W hs ,

Ms∑

i=1

∫

�s,i

whdiv(�hs (v) − v) = 0, (3.26)

∀�i j ∈ �sd ,

∫

�i j

(�h

s (v) − v) · ni j = 0, (3.27)

and there exists a constant C independent of v, h, H, and the diameter of �s,i ,1 ≤ i ≤ Ms, such that

|||�hs (v)|||Xs

≤ C |v|H1(�). (3.28)

Proof The operator �hs is constructed by correcting �h

s :

�hs (v) = �h

s (v) + chs (v)

123


where chs (v)|�s,i ∈ Xh

0,s,i and

∀wh ∈W h0,s,i , 1 ≤ i ≤ Ms,

∫

�s,i

wh div chs (v)=

∫

�s,i

wh div(v − �h

s (v)). (3.29)

Existence of chs (v) follows directly from (3.3) and with the same constant

|chs (v)|H1(�s,i )

≤ 1

β�s‖div(v − �h

s (v))‖L2(�s,i ). (3.30)

The restriction wh ∈ W h0,s,i is relaxed by applying (3.23) and using the fact that ch

s (v)

belongs to Xh0,s,i . Finally, (3.28) follows from the above bound and (3.25).

The idea of constructing the operator �hs via the interior inf-sup condition (3.3)

and the simplified operator �hs satisfying (3.23) and (3.25) is not new. It can be found

for instance in [36] and [12].To control the bilinear form b in �d , we make the following assumption: There

exists a linear operator �hd : H1

0 (�)n �→ V hd satisfying for all v ∈ H1

0 (�)n :

–

∀wh ∈ W hd ,

Md∑

i=1

∫

�d,i

whdiv(�h

d(v) − v) = 0. (3.31)

– For any �i j in �sd ,

∀μH ∈ �Hsd ,

∫

�i j

μH (�hd(v) − �h

s (v)) · ni j = 0. (3.32)

– There exists a constant C independent of v, h, H , and the diameter of �d,i ,1 ≤ i ≤ Md , such that

|||�hd(v)|||Xd

≤ C |v|H1(�). (3.33)

The construction of the operator �hd is presented in Sect. 4. In particular, the general

construction strategy discussed in Sect. 4.1 gives an operator that satisfies (3.31) and(3.32). The stability bound (3.33) is shown to hold for various cases in Sect. 4.4.

The next lemma follows readily from the properties of �hs and �h

d .

123


Lemma 3.6 Under the above assumptions, there exists a linear operator �h ∈L(H1

0 (�)n; V h) such that for all v ∈ H10 (�)n

∀wh ∈ W h,

M∑

i=1

∫

�i

whdiv(�h(v) − v

) = 0, (3.34)

|||�h(v)|||X ≤ C |v|H1(�), (3.35)

with a constant C independent of v, h, H, and the diameter of �i , 1 ≤ i ≤ M.

Proof Take �h(v)|�s = �hs (v) and �h(v)|�d = �h

d(v). Then (3.34) follows from(3.26) and (3.31). The matching condition of the functions of V h at the interfaces of�sd holds by virtue of (3.32). Finally, the stability bound (3.35) stems from (3.28) and(3.33).

The following inf-sup condition between W h and V h is an immediate consequence ofa simple variant of Fortin’s Lemma [18,34] and Lemma 3.6.

Theorem 3.1 Under the above assumptions, there exists a constant β� > 0, indepen-dent of h, H, and the diameter of �i j for all i < j , such that

∀wh ∈ W h, supvh∈V h

b(vh, wh)

|||vh |||X≥ β�‖wh‖L2(�). (3.36)

Finally, well-posedness of the discrete scheme (3.13) follows from Lemma 3.3 or 3.4and Theorem 3.1.

Corollary 3.1 Under the above assumptions, problem (3.13) has a unique solution(uh, ph) ∈ V h × W h and

|||uh |||X + ‖ph‖L2(�) ≤ C(‖ f ‖L2(�) + ‖qd‖L2(�d )

), (3.37)

with a constant C independent of h, H, and the diameter of �i j for all i < j .

Proof Since (3.13) is set into finite dimension, it is sufficient to prove uniqueness,and as uniqueness follows from (3.37), it suffices to prove this stability estimate. Thuslet (uh, ph) solve (3.13), which is a typical linear problem with a non-homogeneousconstraint. By virtue of the discrete inf-sup condition (3.36), there exists a functionuh

q ∈ V h such that

∀wh ∈ W h, b(uhq , wh) = −

∫

�d

wh qd ,

|||uhq |||X ≤ 1

β�‖qd‖L2(�d ). (3.38)

123


Then uh0 = uh − uh

q solves (3.13) with qd = 0 and the coercivity condition (3.15) or(3.18) and the discrete inf-sup condition (3.36) imply that

|||uh0 |||X + ‖ph‖L2(�) ≤ C(‖ f ‖L2(�) + ‖qd‖L2(�d )).

With (3.38), this gives (3.37).

4 Elements of construction of �hs and �h

d

Constructing an operator �hs with values in V h

s , satisfying (3.23)–(3.25), uniformlystable with respect to the diameter of the subdomains and interfaces, is not straight-forward, particularly in 3-D. On the other hand, a general construction of �h

d in �d

can be found in [5], and we shall adapt it so that it matches suitably �hs on �sd . Let

us describe our strategy in each region.

4.1 General construction strategy

Let v ∈ H10 (�)n . In �s , we propose the following three-step construction.

1. Starting step. In each �s,i , 1 ≤ i ≤ Ms , take �hs (v) = Sh(v), where Sh(v) is

a Scott and Zhang [56] approximation operator, constructed so that Sh(v)|∂�s,i

only uses values of v restricted to ∂�s,i , and in particular, Sh(v)|�s = 0. ThusSh(v)|�s,i ∈ Xh

s,i and Sh(v) ∈ Xhs .

2. First correction step. For each �i j ∈ �ss ∪ �sd with i < j , correct �hs (v) in �s,i

by setting

�hs (v)|�s,i := �h

s (v)|�s,i + chi,�i j

(v),

where chi,�i j

(v) ∈ Xhs,i , ch

i,�i j(v) = 0 on ∂�s,i\�i j ,

∫

�i j

chi,�i j

(v) · ni j =∫

�i j

(v − Sh(v)|�s,i

) · ni j , (4.1)

and satisfies suitable bounds. The existence of chi,�i j

(v) (without bounds) is guar-anteed by assumption (3.7). In particular, one can define it on �i j to satisfy (4.1),extend it by zero on ∂�s,i\�i j , and extend it arbitrarily inside �s,i to a function inXh

s,i . Note that the correction chi,�i j

(v) only affects values at points located in theinterior of �s,i and in the interior of �i j . It has no influence on values at pointslocated on the other interfaces. For this reason, the discrete term that appears inthe right-hand side of (4.1) is the trace on �i j of Sh(v) coming from �s,i ; thisterm has not been yet corrected. Once this correction step is performed on all�i j ∈ �ss ∪ �sd with i < j , the resulting function �h

s (v) satisfies on all theseinterfaces

123


∫

�i j

(�h

s (v)|�s,i − v) · ni j = 0. (4.2)

Note that this step does not modify the trace on �i j of Sh(v) coming from �s, j .This trace will be modified in the next step.

3. Second correction step. For each �i j ∈ �ss with i < j , correct �hs (v) in �s, j by

setting

�hs (v)|�s, j := �h

s (v)|�s, j + chj,�i j

(v),

where chj,�i j

(v) ∈ Xhs, j , ch

j,�i j(v) = 0 on ∂�s, j\�i j ,

∀μH ∈ �Hs ,

∫

�i j

μH · chj,�i j

(v) =∫

�i j

μH · (�hs (v)|�s,i − Sh(v)|�s, j

), (4.3)

and satisfies suitable bounds. The existence of chj,�i j

(v) (without bounds) is guar-anteed by assumption (3.8). In particular, one can define it on �i j to satisfy (4.3),extend it by zero on ∂�s, j\�i j , and extend it arbitrarily inside �s, j to a func-tion in Xh

s, j . Here also, the correction chj,�i j

(v) has no influence on values atpoints located on the other interfaces. Once this correction step is performed on all�i j ∈ �ss with i < j , the resulting function �h

s (v) satisfies on all these interfacesbs(�

hs (v),μH ) = 0 for all μH ∈ �H

s , i.e.

∀μH ∈ �Hs ,

∫

�i j

[�hs (v)] · μH = 0. (4.4)

Finally, if the second assumption in Hypothesis 3.2 holds, then n ∈ �Hs and (4.2)

and (4.4) imply that on all these interfaces,

∫

�i j

(�h

s (v)|�s, j − v) · n = 0. (4.5)

Therefore �hs (v) satisfies (3.23) and (3.24). Specific constructions of the cor-

rections chi,�i j

(v) and chj,�i j

(v) that guarantee that �hs (v) also satisfies (3.25) are

presented in the forthcoming subsections.

Next, we propose the following two-step construction algorithm in �d .

1. Starting step. Set Phd (v) = Rh(v) ∈ Xh

d , where Rh(v) is a standard mixed approx-imation operator associated with W h

d . It preserves the normal component on theboundary:

123


∀�i j ⊂ ∂�d,k, 1≤k ≤ Md , ∀vh ∈ Xhd ,

∫

�i j

vh · ni j(Rh(v)|�d,k − v

) · ni j = 0,

(4.6)

and satisfies

∀1 ≤ i ≤ Md , ∀wh ∈ W hd ,

∫

�d,i

whdiv(Rh(v) − v

) = 0. (4.7)

2. Correction step. It remains to prescribe the jump condition. For each �i j ∈ �dd ∪�sd with i < j , correct Ph

d (v) in �d, j by setting:

Phd (v)|�d, j := Ph

d (v)|�d, j + chj,�i j

(v),

where chj,�i j

(v) ∈ Xhd, j , ch

j,�i j(v) · n j = 0 on ∂�d, j\�i j , div ch

j,�i j(v) = 0 in

�d, j ,

∀μH ∈ �Hd ,

∫

�i j

μH chj,�i j

(v) · ni j =∫

�i j

μH (Rh(v)|�d,i − Rh(v)|�d, j

) · ni j ,

∀μH ∈ �Hsd ,

∫

�i j

μH chj,�i j

(v) · ni j =∫

�i j

μH (�hs (v)|�s,i − Rh(v)|�d, j

) · ni j ,

(4.8)

and chj,�i j

(v) satisfies adequate bounds. Existence of a non necessarily divergence-

free chj,�i j

(v) (without bounds) follows from (3.9); it suffices to extend suitably

Rh(v)|�d, j and Rh(v)|�d,i or �hs (v)|�s,i . The zero divergence will be prescribed

in the examples. Note that chj,�i j

(v) has no effect on interfaces other than �i j and

no effect on the restriction of Phd (v) in �d,i or on that of �h

s (v) in �s,i . Thereforethese corrections can be superimposed.

When step 2 is done on all �i j ∈ �dd ∪ �sd with i < j , the resulting function Phd (v)

has zero normal trace on �d , it belongs to V hd since, due to the first equation in (4.8),

it satisfies for all �i j ∈ �dd with i < j

∀μH ∈ �Hd ,

∫

�i j

μH [Phd (v) · n] = 0, (4.9)

and, as the corrections are assumed to be divergence-free in each subdomain,

∀wh ∈ W hd , ∀1 ≤ i ≤ Md ,

∫

�d,i

whdiv(Ph

d (v) − v) = 0. (4.10)

123


Furthermore, due to the second equation in (4.8), it satisfies for all �i j ∈ �sd ,

∀μH ∈ �Hsd ,

∫

�i j

μH (�hs (v)|�s,i − Ph

d (v)|�d, j

) · ni j = 0. (4.11)

Therefore, taking �hd(v) = Ph

d (v) in �d , it satisfies (3.31) and (3.32).The remainder of this section is devoted to corrections ch

i,�i j(v) and ch

j,�i j(v) where

the constants in the stability bounds (3.25) and (3.33) are shown to be independentof the discretization parameters and the diameter of the subdomains. In particular,the bounds for the corrections will stem from the fact that each involves differencesbetween v and good approximations of v. Beforehand, we need to refine the assump-tions on the meshes at the interfaces and refine Hypothesis 3.1 on the mesh of subdo-mains.

Hypothesis 4.1 For i < j , let T be any element of T hi that is adjacent to �i j , and let

{T�} denote the set of elements of T hj that intersect T . The number of elements in this

set is bounded by a fixed integer and there exists a constant C such that

|T�| |T |−1 ≤ C.

The same is true if the indices i and j of the triangulations are interchanged. Theseconstants are independent of i , j , h, and the diameters of the interfaces and subdo-mains.

Hypothesis 4.2 1. Each �i , 1 ≤ i ≤ M, is the image of a “reference” polygonal orpolyhedral domain by an homothety and a rigid body motion:

�i = Fi (�i ), x = Fi (x) = Ai Ri x + bi , (4.12)

where Ai = diam(�i ), Ri is an orthogonal matrix with constant coefficients andbi a constant vector.

2. There exists a constant σ1 independent of M such that for any pair of adjacentsubdomains �i and � j , 1 ≤ i, j ≤ M, we have

Ai A−1j ≤ σ1. (4.13)

By (4.12) diam(�i ) = 1. In addition, it follows from Hypothesis 3.1 that on onehand the reference domains �i can take at most L geometrical shapes and on the otherhand,

∀i, 1 ≤ i ≤ M, diam(Bi ) ≥ 1

σ, (4.14)

where Bi is the largest ball contained in �i and σ is the constant of (3.1).

123


4.2 A construction of chi,�i j

(v)

In this section we construct a correction chi,�i j

(v) satisfying (4.1) and suitable boundsneeded to establish the stability estimate (3.25). Recall the approximation propertiesof the Scott and Zhang operator of degree r ≥ 1. Let v ∈ Ht (�s,i ), for some realnumber t ≥ 1 and let T be an element of T h

i , 1 ≤ i ≤ Ms . Let �T denote the macro-element that is used for defining the values of Sh(v) in T . Then (cf. [56]) there existsa constant C that depends only on r , t , and the shape regularity of T h

i , such that thefollowing local approximation error holds

‖v − Sh(v)‖L2(T ) + hT |v − Sh(v)|H1(T ) ≤ Chmin(r,t)T |v|Hmin(r,t)(�T ). (4.15)

Owing to the regularity of T hi , when (4.15) is summed over all T in T h

i , it gives

‖v − Sh(v)‖L2(�s,i )+ h|v − Sh(v)|H1(�s,i )

≤ Chmin(r,t)|v|Hmin(r,t)(�s,i ). (4.16)

Let v ∈ H10 (�)n . Consider the case when n = 3, the case n = 2 being simpler, and

also consider the case when �i j is a polyhedral surface, not necessarily a flat plane. LetXh

i,�i jdenote the trace of Xh

s,i on �i j , and T hi,�i j

the trace of T hi on �i j , 1 ≤ i ≤ Ms ,

i < j ; T hi,�i j

is a triangular mesh of each flat face in �i j . In all cases considered, the

restriction to each T of functions of Xhs,i contains at least Pn

1. Now, choose one of

the faces, say F , of �i j . For each interior vertex ak of T hi,�i j

, 1 ≤ k ≤ NF , on F , let

Ok denote the macro-element of all triangles of T hi,�i j

(i.e. faces on F) that share thevertex ak . Thus

F = ∪NFk=1Ok .

The set {Ok} is not a partition of F , but it can be transformed into a partition by setting

�1 = O1, and recursively �k = Ok\ ∪k−1�=1 ��.

Note that some �k may be empty. By construction, we have

F = ∪NFk=1�k, �k ∩ �� = ∅, k �= �, �k ⊂ Ok .

This can be done for all flat faces of �i j . For each k, let bk be the piecewiseP1 “bubble”function such that

bk(a�) = δk,�,

123


extended by zero to all vertices inside �s,i , and define chi,�i j

(v) by

chi,�i j

(v)=∑

F⊂�i j

NF∑

k=1

ckbk, where ck = 1∫Ok

bk

∫

�k

(v−Sh(v)|�s,i

), ck =0, if �k =∅.

(4.17)

Lemma 4.1 The correction chi,�i j

(v) defined by (4.17) satisfies (4.1), more precisely

∀F ⊂ �i j ,

∫

F

chi,�i j

(v) =∫

F

(v − Sh(v)|�s,i

), (4.18)

and there exists a constant C independent of h and the diameter of �s,i and �i j suchthat

∀v ∈ H10 (�)n, |ch

i,�i j(v)|H1(�s,i )

≤ C |v|H1(�s,i ), ‖ch

i,�i j(v)‖L2(�s,i )

≤ Ch|v|H1(�s,i ).

(4.19)

Proof For each k, the support of the trace of bk on �i j is contained in a single flatplane, say F , therefore,

∫

F

chi,�i j

(v) =∫

F

NF∑

k=1

ckbk =NF∑

k=1

ck

∫

F

bk =NF∑

k=1

ck

∫

Ok

bk =NF∑

k=1

∫

�k

(v − Sh(v)|�s,i

)

=∫

F

(v − Sh(v)|�s,i

),

since the set (�k) is a partition of �.To derive the estimate (4.19), we consider the faces T ′ in �k , pass to the reference

element T , where T is the element of T hi adjacent to T ′, apply a trace theorem in T ,

and revert to T . This gives

∣∣∣∣∣∣∣

∫

�k

(v−Sh(v)|�s,i

)

∣∣∣∣∣∣∣≤∑

T ′⊂�k

‖v − Sh(v)|�s,i ‖L1(T ′)

≤ C∑

T ′⊂�k

|T ′| |T |− 12

(‖v−Sh(v)‖L2(T )+hT |v−Sh(v)|H1(T )

).

In this proof, C denotes constants that depend only on the reference element and theshape regularity of the triangulation. Then considering the regularity of T h

i , the localapproximation formula (4.15) with r = t = 1 implies that

123


∣∣∣∣∣∣∣

∫

�k

(v − Sh(v)|�s,i

)

∣∣∣∣∣∣∣≤ C

∑

T ′⊂�k

hT |T ′| |T |− 12 |v|H1(�T ) ≤ C |�k | ρ− 1

2k |v|H1(Dk )

,

(4.20)

where Dk is the set of elements of T hi where the values of v are taken for computing

Sh(v), and

ρk = minT ∈Dk ρT .

Therefore, considering that |�k | ≤ |Ok |, ck defined by (4.17) satisfies

|ck | ≤ C ρ− 1

2k |v|H1(Dk )

. (4.21)

Now,

|chi,�i j

(v)|2H1(�s,i )=∫

�s,i

∣∣∣∣∣∣

∑

F⊂�i j

NF∑

k=1

ck∇ bk

∣∣∣∣∣∣

2

=∑

T ∈T hi

∫

T

∣∣∣∣∣∣

∑

F⊂�i j

NF∑

k=1

ck∇ bk

∣∣∣∣∣∣

2

.

But given an element T in T hi there is at most a fixed (and small) number of indices

k such that bk |T �= 0. Therefore

|chi,�i j

(v)|2H1(T )=∫

T

∣∣∣∣∣∣

∑

F⊂�i j

NF∑

k=1

ck∇ bk

∣∣∣∣∣∣

2

≤ C∑

k

|ck |2|bk |2H1(T ),

where the sum runs over all indices k such that bk |T �= 0. By substituting (4.21) intothis inequality and estimating the norm of bk , we easily derive that

|chi,�i j

(v)|2H1(T )≤ C

∑

k

1

ρk

1

ρ2T

|T ||v|2H1(Dk ). (4.22)

Since ρkρ2T has the same order as |T |, then by summing (4.22) over all T in T h

i andconsidering that there is at most a fixed (and small) number of repetitions in the Dk ,we obtain the first inequality in (4.19). The second inequality in (4.19) follows in asimilar manner from the representation (4.17) and bound (4.21).

4.3 A construction of chj,�i j

(v) in �s . The 2-D case

Constructing a suitable correction solving (4.3) is fairly complex because it amountsto satisfying many conditions per interface. In 2-D, it can be done by following thesharp approach of Crouzeix and Thomée [23]. This is a general procedure that can

123


also be used in 3-D, but as observed in Remark 4.1, it restricts the meshes. To bypassthis restriction, we present a more ad hoc construction in 3-D, which is postponed tothe Appendix: Sect. 7.2.

We slightly modify the underlying Scott and Zhang operator by asking that themodified operator Sh be continuous at the end points of all interfaces �i j in �ss ∪�sd ,and coincide elsewhere with Sh . This only concerns the set of points, say ak , 1 ≤ k ≤P , that do not lie on �s , since Sh(v) is necessarily zero on �s . Note that these are allinterior cross points of the triangulation of subdomains T�. To achieve continuity atany such point a, we pick an element Ta in �s with vertex a and set

Sh(v)(a) = 1

|Ta|∫

Ta

v.

Both corrections chi,�i j

(v) and chj,�i j

(v) are defined to satisfy respectively (4.1)

and (4.3) with Sh(v) replaced by Sh(v). For v ∈ H1(�), this does not change theapproximation properties (4.16) of Sh and hence (4.19), except that the domain ofdefinition of v is possibly a little larger than �s,i or �s, j near the end points of �i j .

Notation 4.1 From now on, we denote by �s,k the possibly slightly enlarged domainof definition of v in case Sh(v) is used, with the convention that it coincides with �s,k

when Sh(v) is not used.

Owing to the continuity of Sh(v) at the end points of �i j and the fact that all previouslymade corrections vanish at the end points of the interfaces �i j in �ss∪�sd , the integrandin the right-hand side of (4.3) also vanishes at the end points of �i j . Thus the trace of

�hs (v)|�s,i − Sh(v)|�s, j on �i j belongs to H

12

00(�i j )2, see Sect. 1.1. We shall see that

this property is crucial for estimating uniformly the gradient of chj,�i j

(v).

We propose to split the construction of chj,�i j

(v) into two parts:

1. Construct an operator πhs ∈ L(H

12

00(�i j )2, Xh

j,�i j) such that for all � ∈ H

12

00(�i j )2,

∀μH ∈ �Hs ,

∫

�i j

(πhs (�) − �) · μH = 0,

πhs (�) = 0, at the end points of �i j , (4.23)

|πhs (�)|

H12

00 (�i j )≤ κs |�|

H12

00 (�i j ),

with a constant κs that is independent of h, H , and the measure of �i j .2. Define ch

j,�i j(v) to be a suitable extension of πh

s (�hs (v)|�s,i − Sh(v)|�s, j ) inside

�s, j so that it satisfies all uniform properties required for the stability bound (3.25).

Note that chj,�i j

(v) satisfies (4.3) by construction. Also, the existence of the operator

πhs implies that condition (3.8) holds.

123


We now discuss the construction of an operator satisfying (4.23). Uniformity withrespect to the diameters of �i j and �s, j is obtained by reverting to the referencesubdomains. Let � j be associated with �s, j by Hypothesis 4.2 and let � = F−1

j (�i j ).

Let the image by F−1j of T h

j,�i jbe denoted by T j . Similarly, set

X j = {v = vh ◦ F−1j ; vh ∈ Xh

j,�i j, vh = 0, at the end points of �i j },

� = {μ = μH ◦ F−1j ; μH ∈ �H

s }.

Then, given � in H12

00(�)2, if we construct π j (�) in X j unique solution of

∀μ ∈ �,

∫

�

π j (�) · μ =∫

�

� · μ, (4.24)

and satisfying

|π j (�)|H

12

00 (�)< C |�|

H12

00 (�), (4.25)

with a constant C independent of j and �, then by reverting to �s, j and defining πhj (�)

by

πhj (�) ◦ Fj = π j (� ◦ Fj ) = π j (�), (4.26)

we shall obtain an operator satisfying (4.23). Indeed, uniqueness of the solution of(4.24) will guarantee that the mapping π j is linear, and the inequality in (4.23) with the

same constant will follow from the invariance of the H12

00 seminorm by a homothetyand a rigid body motion. But since explicit constructions of π j are fairly technical,we postpone them to the Appendix: Sect. 7.1 and discuss the extension part (2) now.First we construct a suitable function v in H1(�s, j ).

Lemma 4.2 For any �s, j ⊂ �s , any interface � ⊂ ∂�s, j and any � ∈ H12

00(�) thereexists a function v = E(�) ∈ H1(�s, j ) satisfying

v|� = �, v|∂�s, j \� = 0,

the mapping E is linear and there exists a constant C independent of �, v, and thediameter of � and �s, j such that

|v|H1(�s, j )≤ C |�|

H12

00 (�), ‖v‖L2(�s, j )

≤ C A j |�|H

12

00 (�), (4.27)

where A j is the diameter of �s, j .

123


Proof With the notation of Hypothesis 4.2, let � = � ◦ Fj . Then � belongs to H12

00(�);hence it can be extended by zero to ∂� j and the extended function, say �0 belongs to

H12 (∂� j ) with

|�0|H

12 (∂� j )

≤ C |�|H

12

00 (�). (4.28)

There exists an extension operator E ∈ L(H12 (∂� j ); H1(� j )) such that

‖E‖L(H

12 (∂� j );H1(� j ))

≤ C . (4.29)

Take v = E(�0) and v = v◦F−1j . The mapping � �→ v is linear, v belongs to H1(�s, j ),

its trace satisfies the statement of the lemma, and it remains to check (4.27). Thefollowing inequalities stem from (4.29), (4.28) and a straightforward scaling argument:

|v|H1(�s, j )≤ |v|H1(� j )

≤ C |�0|H

12 (∂� j )

≤ C |�|H

12

00 (�)≤ C |�|

H12

00 (�),

‖v‖L2(�s, j )≤ C A j‖v‖L2(� j )

≤ C A j |�|H

12

00 (�).

(4.30)

Next, we take � = �i j and define

chj,�i j

(v) = Sh(w), w = E(πhs (�h

s (v)|�s,i − Sh(v)|�s, j )), (4.31)

where the Scott and Zhang interpolant Sh is constructed so that Sh(w) vanishes on∂�s, j\�i j . The uniform approximation properties (4.16) of Sh , (4.25), and (4.27)imply, with constants C independent of h and the diameters of �i j , �s,i and �s, j :

‖chj,�i j

(v)‖H1(�s, j )≤C |w|H1(�s, j )

≤C∣∣{�h

s (v)|�s,i − Sh(v)|�s, j }∣∣

H12

00 (�i j ), (4.32)

and it remains to derive a uniform bound for the last norm in terms of v. This is theobject of the next lemma.

Lemma 4.3 Let �i j ∈ �ss , i < j . There exists a constant C independent of h and thediameters of �i j , �s,i and �s, j , such that

∀v ∈ H10 (�)2,

∣∣{�hs (v)|�s,i − Sh(v)|�s, j }

∣∣H

12

00 (�i j )≤C |v|H1(�s,i ∪�s, j )

, (4.33)

where �s,k is defined in Notation 4.1.

123


Proof Recall that the trace on �i j of �hs (v)|�s,i is Sh(v)|�s,i + ch

i,�i j(v). As ch

i,�i j(v)

vanishes on ∂�s,i\�i j , then by passing to the reference subdomain �i , we obtain

|chi,�i j

(v)|H

12

00 (�i j )= |ch

i,�i j(v) ◦ Fi |

H12

00 (�)

≤ C |chi,�i j

(v) ◦ Fi |H1(�i )≤ C |ch

i,�i j(v)|H1(�s,i )

.

Therefore it is bounded owing to (4.19), with �s,i instead of �s,i . Thus it suffices toconsider the jump [Sh(v)] through �i j .

First, by writing [Sh(v)] = [Sh(v) − v], by passing to the reference subdomains�k , k = i, j , and by using the approximation properties (4.16) of Sh , we derive

∣∣(Sh(v) − v

)|�s,k

∣∣H

12 (�i j )

= ∣∣{(Sh(v)−v) ◦ Fk |�k

}∣∣H

12 (�)

≤ C‖(Sh(v)−v) ◦ Fk‖H1(�k)

≤ C(|Sh(v) − v|2H1(�s,k)

+ A−2k ‖Sh(v) − v‖2

L2(�s,k)

) 12

≤ C(|v|2

H1(�s,k)+ (h A−1

k )2|v|2H1(�s,k )

) 12. (4.34)

Since h < Ak , this bounds the first part of the norm and it remains to estimate

∫

�i j

1

d(x)

∣∣[Sh(v)]∣∣2,

where d(x) denotes the distance between x and the end points of �i j . This part doesnot have the same immediate bound as the above correction because the jump doesnot vanish on the other boundaries of the subdomains. Let us first consider the casewhen �i j is a straight line. There is no loss of generality in assuming that �i j lies onthe axis y = 0 with end point at the origin: �i j =]0, L[. Let 0 = x0 < x1 < · · · <

xN < xN+1 = L denote the nodes of T hi,�i j

and exceptionally set hn = xn+1 − xn .Then d(x) = Min(x, L − x) and since

∫

�i j

1

d(x)

∣∣[Sh(v)]∣∣2 ≤

L∫

0

1

x

∣∣[Sh(v)]∣∣2 +

L∫

0

1

L − x

∣∣[Sh(v)]∣∣2,

it suffices to examine the first term:

L∫

0

1

x

∣∣[Sh(v)]∣∣2 =N∑

n=0

xn+1∫

xn

1

x

∣∣[Sh(v)]∣∣2.

For any n ≥ 1, we have xn ≥ hn−1, and since [Sh(v)] belongs to(H1(0, L) ∩

C0(0, L))2, we can write

123


xn+1∫

xn

1

x

∣∣[Sh(v)]∣∣2 ≤ 1

xn

xn+1∫

xn

⎛

⎝[Sh(v)(xn)] +x∫

xn

d

dt[Sh(v)(t)]dt

⎞

⎠

2

dx

≤ 2hn

hn−1

(

[Sh(v)(xn)]2 + hn

2

∥∥∥∥

d

dx[Sh(v)]

∥∥∥∥

2

L2(xn ,xn+1)

)

. (4.35)

An equivalence of norms yields

2hn

hn−1

∣∣[Sh(v)(xn)]∣∣2 ≤ 2hn

hn−1‖[Sh(v)]‖2

L∞(xn ,xn+1)≤ C

1

hn−1‖[Sh(v)]‖2

L2(xn ,xn+1)

= C1

hn−1‖(Sh(v)|�s,i − v) − (Sh(v)|�s, j − v)‖2

L2(xn ,xn+1).

By arguing as in the proof of Lemma 4.1, we obtain

‖Sh(v)|�s,i − v‖L2(xn ,xn+1)≤ Ch

12T |v|H1(�n,i )

, (4.36)

where T is the element of T hi adjacent to [xn, xn+1] and �n,k denotes the set of

elements in �s,k used in defining Sh(v)|�s,k on [xn, xn+1].The estimation of the second term is more technical because now Sh(v) is con-

structed on T hj whereas [xn, xn+1] is the side of an element of T h

i . Let {T�} denote the

set of elements of T hj that intersect [xn, xn+1]. The argument of Lemma 4.1 gives:

‖Sh(v)|�s, j − v‖L2(xn ,xn+1)≤ C

∑

�

h12T�

|v|H1(�n,�). (4.37)

Then combining (4.36) and (4.37), using the regularity of the triangulation and Hypoth-esis 4.1, we obtain

2hn

hn−1

∣∣[Sh(v)(xn)]∣∣2 ≤ C(|v|2

H1(�n,i )+ |v|2

H1(�n, j )

). (4.38)

Similarly,

h2n

hn−1

∥∥∥d

dx[Sh(v)]

∥∥∥2

L2(xn ,xn+1)≤ C

1

hn−1‖[Sh(v)]‖2

L2(xn ,xn+1)

= C1

hn−1‖[Sh(v) − v]‖2

L2(xn ,xn+1)

≤ C(|v|2

H1(�n,i )+ |v|2

H1(�n, j )

). (4.39)

When n = 0, as [Sh(v)(0)] = 0, there only remains the second term in the first lineof (4.35), and we immediately derive

123


x1∫

0

1

x

∣∣[Sh(v)]∣∣2 =x1∫

0

1

x

( x∫

0

d

dt

([Sh(v)])dt)2

dx

≤x1∫

0

∥∥∥d

dx

([Sh(v)])∥∥∥

2

L2(0,x)≤ h1

∥∥∥d

dx

([Sh(v)])∥∥∥

2

L2(0,x1)

≤ C(|v|2

H1(�0,i )+ |v|2

H1(�0, j )

).

Finally, when �i j is a polygonal line, the above argument is valid on the segments thatshare an end point with �i j and is simpler on the other segments as d(x) is not smallcompared to h.

Inequalities (4.32) and (4.33) imply the following.

Corollary 4.1 Let �i j ∈ �ss , i < j . There exists a constant C independent of h andthe diameters of �i j , �s,i and �s, j , such that

∀v ∈ H10 (�)2, ‖ch

j,�i j(v)‖H1(�s, j )

≤ C |v|H1(�s,i ∪�s, j ). (4.40)

Corollary 4.2 The approximation operator �hs constructed in Sect. 4.1 with the above

corrections chi,�i j

(v) and chj,�i j

(v) satisfies assumption (3.25).

Proof Since �hs (v) is constructed by correcting the Scott-Zhang interpolant Sh with

chi,�i j

(v) and chj,�i j

(v), assumption (3.25) follows from (4.16), (4.19), (4.40), and thePoincaré inequality (1.1).

Remark 4.1 The above correction chj,�i j

(v) has a straightforward extension in 3-D,

but its use is limited because now it requires continuity of Sh(v) on the edges ofsubdomains; thus the meshes must match on these edges. This is not required by thecorrection defined in Sect. 7.2.

4.4 A construction of chj,�i j

(v) in �d

Here we construct a correction chj,�i j

(v) in �d satisfying (4.8) and suitable continuitybounds that are needed to establish the stability estimate (3.33). Recall that the exis-tence of ch

j,�i j(v) relies on (3.9). In the construction below we directly show that (3.9)

holds for a wide range of mesh configurations.Let v be given in H1

0 (�)n . Recall that the mixed approximation operator Rh definedin each �d,i takes its values in Xh

d and satisfies (4.6) on each �i j ⊂ ∂�d,k , 1 ≤ k ≤Md , and (4.7) in each �d,i , 1 ≤ i ≤ Md :

123


∀vh ∈ Xhd ,

∫

�i j

vh · ni j(Rh(v)|�d,k − v

) · ni j = 0,

∀wh ∈ W hd ,

∫

�d,i

whdiv(Rh(v) − v

) = 0.

Furthermore there exists a constant C independent of h and the geometry of �d,i , suchthat

∀v ∈ H10 (�)n, ‖Rh(v)‖H(div;�d,i ) ≤ C‖v‖H1(�d,i )

, 1 ≤ i ≤ Md . (4.41)

This is easily established by observing that the moments defining the degrees of free-dom of Rh(v) are invariant by homothety and rigid-body motion; in particular thenormal vector is preserved. In addition, it satisfies (3.6):

∀i, 1 ≤ i ≤ Md , ∀vh ∈ Xhd,i , div vh ∈ W h

d,i .

The above properties also imply (3.5): for all i , 1 ≤ i ≤ Md ,

infwh∈W h

0,d,i

supvh∈Xh

0,d,i

∫�d,i

whdiv vh

‖vh‖H(div;�d,i )‖wh‖L2(�d,i )

≥ β�d ,

with a constant β�d > 0 independent of h and Ai .

Now, let �i j ∈ �dd ∪ �sd ; by analogy with the situation in the Stokes region, wedenote by Xh

d, j,�i jthe trace space of Xh

d, j on �i j . Following [5], we define the spaceof normal traces

X nj,�i j

= {w · ni j ; w ∈ Xhd, j,�i j

},

and the orthogonal projection Qhj,�i j

from L2(�i j ) into X nj,�i j

. Then we make thefollowing assumption: There exists a constant C , independent of H , h, i , j , and thediameters of �i j and �d, j , such that

∀μH ∈ �Hd ,∀μH ∈ �H

sd , ‖μH ‖L2(�i j )≤ C‖Qh

j,�i j(μH )‖L2(�i j )

. (4.42)

It is shown in [60] that (4.42) holds for both continuous and discontinuous mortarspaces, if the mortar grid T H

i j is a coarsening by two of T hj,�i j

. A similar inequalityfor more general grid configurations is shown in [52]. Formula (4.42) implies that theprojection Qh

j,�i jis an isomorphism from the restriction of �H

sd , respectively �Hd , to

�i j , say �Hsd,i j respectively �H

d,i j , onto its image in X nj,�i j

, and the norm of its inverseis bounded by C . Then a standard algebraic argument shows that its dual operator,namely the orthogonal projection from X n

j,�i jinto �H

sd,i j , respectively �Hd,i j , is also

an isomorphism from the orthogonal complement in X nj,�i j

of the projection’s kernel

123


onto �Hsd,i j , respectively �H

d,i j , and the norm of its inverse is also bounded by C . As

a consequence, for each f ∈ L2(�i j ), there exists vh · ni j ∈ X nj,�i j

such that

∀μH ∈ �Hd ,∀μH ∈ �H

sd ,

∫

�i j

μH vh · ni j =∫

�i j

f μH ,

and there exists a constant C independent of h, and the diameter of �i j , such that

‖vh · ni j‖L2(�i j )≤ C‖ f ‖L2(�i j )

.

This implies that (3.9) holds. Furthermore, the solution vh ·ni j is unique in the orthog-onal complement of the projection’s kernel and by virtue of this uniqueness, vh · ni j

depends linearly on f . This result permits to partially solve (4.8).

Lemma 4.4 Let v ∈ H10 (�)n. Under assumption (4.42), for each �i j ∈ �dd ∪ �sd ,

there exists wh · ni j ∈ X nj,�i j

such that

∀μH ∈ �Hd ,

∫

�i j

μH wh · ni j =∫

�i j

μH [Rh(v) · n],

‖wh · ni j‖L2(�i j )≤ C‖[Rh(v) · n

]‖L2(�i j ),

(4.43)

∀μH ∈ �Hsd ,

∫

�i j

μH wh · ni j =∫

�i j

μH (�hs (v)|�s,i − Rh(v)|�d, j

) · ni j ,

‖wh · ni j‖L2(�i j )≤ C‖(�h

s (v)|�s,i − Rh(v)|�d, j

) · ni j‖L2(�i j ),

(4.44)

with the constant C of (4.42). The mapping v �→ wh · ni j is linear.

Lemma 4.4 constructs a normal trace wh · ni j on �i j and we must extend it inside�d, j . To this end, let �h ∈ L2(∂�d, j ) be the extension of wh · ni j by zero on ∂�d, j .Next, we solve the problem: Find q ∈ H1(�d, j ) ∩ L2

0(�d, j ) such that

� q = 0 in �d, j ,∂q

∂n j= �h on ∂�d, j . (4.45)

Lemma 4.5 Problem (4.45) has one and only one solution q ∈ H32 (�d, j )∩L2

0(�d, j )

and

|q|H1(�d, j )≤ C

√A j‖[Rh(v) · n]‖L2(�i j )

,

|q|H

32 (�d, j )

≤ C‖[Rh(v) · n]‖L2(�i j ), �i j ∈ �dd , (4.46)

|q|H1(�d, j )≤ C

√A j‖(�h

s (v)|�s,i − Rh(v)|�d, j

) · ni j‖L2(�i j ),

123


|q|H

32 (�d, j )

≤ C‖(�hs (v)|�s,i − Rh(v)|�d, j

) · ni j‖L2(�i j ), �i j ∈ �sd , (4.47)

with constants C independent of h, H, q, i , j , and the diameters of �i j and �d, j .

Proof Since μH contains the constant functions, �h satisfies on �i j ∈ �dd , using(4.43) and (4.6),

∫

∂�d, j

�h =∫

�i j

wh · ni j =∫

�i j

[(Rh(v) − v) · n] = 0,

and this implies the unique solvability of (4.45). A similar argument holds on �i j ∈ �sd

using (4.44) and (3.24). In order to check (4.46), we pass to the reference subdomain� j associated with �d, j , let n j be its exterior unit normal vector, � = F−1

j (�i j ),

q = q ◦ Fj and � = �h ◦ Fj = (wh ◦ Fj ) · n j ; then q ∈ H1(� j ) ∩ L20(� j ) is the

unique solution of

� q = 0 in � j ,∂q

∂ n j= A j � on ∂� j .

As � is in L2(∂� j ), it follows from [43] that q belongs to H32 (� j ) and

‖q‖H

32 (� j )

≤ C A j‖�‖L2(∂� j ),

with a constant C that only depends on the geometry of � j . Then (4.46) is a directconsequence of (4.43) and the following bounds:

‖�h‖L2(�i j )= A

n−12

j ‖�‖L2(∂� j ), |q|H1(�d, j )

= An2 −1j |q|H1(� j )

,

|q|H

32 (�d, j )

= An−3

2j |q|

H32 (� j )

.

The argument for (4.47) is similar, using (4.44).

Now define c = ∇ q in �d, j . Then c belongs to H(div;�d, j ) ∩ H12 (�d, j )

n anddiv c = 0. Therefore Rh(c) is well defined [18] and satisfies the approximation prop-erty for divergence-free functions [48]

‖c − Rh(c)‖L2(�d, j )≤ Chr |c|Hr (�d, j ), 0 < r ≤ 1

2, (4.48)

with a constant C independent of h, j , and the diameter of �d, j . We are now ready todefine the correction ch

j,�i j(v). In particular, take ch

j,�i j(v) = Rh(c) applied in �d, j .

Note that chj,�i j

(v) belongs to Xhd, j , and (4.7) and (4.6) imply that div ch

j,�i j(v) = 0

123


in �d, j and chj,�i j

(v) · n j = �h = wh · ni j on �i j . Therefore (4.43) and (4.44) imply

that chj,�i j

(v) satisfies (4.8). Furthermore, (4.48) yields

‖chj,�i j

(v)‖L2(�d, j )≤ ‖Rh(c) − c‖L2(�d, j )

+ ‖c‖L2(�d, j )

≤ Ch12j |c|

H12 (�d, j )

+‖c‖L2(�d, j ),

with a constant C independent of the geometry of �d, j . Considering that h j ≤ A j ,Lemma 4.5 gives

‖chj,�i j

(v)‖L2(�d, j )≤ C

√A j‖[Rh(v) · n]‖L2(�i j )

, �i j ∈ �dd , (4.49)

‖chj,�i j

(v)‖L2(�d, j )≤ C

√A j‖(�h

s (v)|�s,i − Rh(v)|�d, j ) · ni j‖L2(�i j ), �i j ∈ �sd ,

(4.50)

with a constant C independent of h, H , v, i , j , and the diameters of �i j and �d, j .

Corollary 4.3 The approximation operator �hd constructed in Sect. 4.1 with correc-

tions chj,�i j

(v) described above satisfies assumption (3.33).

Proof Since �hd is a correction of the mixed interpolant Rh , which is stable in the sense

of (4.41), it remains to bound ‖chj,�i j

(v)‖H(div;�d, j ). By construction div chj,�i j

(v) = 0.In the following we will make use of the trace inequality [38]

∀ϕ ∈ H1(�d, j ), ‖ϕ‖L2(�i j )≤ C

(A

− 12

j ‖ϕ‖L2(�d, j )+ A

12j |ϕ|H1(�d, j )

). (4.51)

For �i j ∈ �dd , using (4.49), (4.6), and (4.51), we have

‖chj,�i j

(v)‖L2(�d, j )≤ C

√A j (‖v · ni‖L2(�i j )

+ ‖v · n j‖L2(�i j )) ≤ C‖v‖H1(�d,i ∪�d, j )

,

using also that A j ≤ 1. For �i j ∈ �sd , we employ (4.50) and (4.1) to obtain

‖chj,�i j

(v)‖L2(�d, j )≤ C

√A j(‖(v − Rh(v)|�d, j ) · n j‖L2(�i j )

+‖(v − (Sh(v) + chi,�i j

(v))|�s,i ) · ni‖L2(�i j )

),

with chi,�i j

(v) defined in (4.1) and constructed in Sect. 4.2, and if necessary Sh replaced

by Sh . For the first term in the right-hand side, using (4.6), we have

‖(v − Rh(v)|�d, j ) · n j‖L2(�i j )≤ ‖v · n j‖L2(�i j )

≤ C‖v‖H1(�d, j ).

For the second term on the right, using (4.51) and the approximation property (4.16)of Sh or Sh we have

123


‖(v − Sh(v)|�s,i ) · ni‖L2(�i j )≤ C(A

− 12

i ‖v−Sh(v)‖L2(�s,i )+ A

12i |v − Sh(v)|H1(�s,i )

)

≤ C(A− 1

2i hi |v|H1(�s,i )

+ A12i |v|H1(�s,i )

)

≤ C |v|H1(�s,i ),

using that hi < Ai ≤ 1, with a similar bound for ‖chi,�i j

(v)‖L2(�s,i ), in view of (4.19).

The proof is completed by combining all bounds and using the Poincaré inequality(1.1).

5 Error analysis

In this section we establish a priori error estimates for our method. Let us assumethat the finite element spaces Xh

s and W hs in �s contain at least polynomials of degree

rs and rs − 1, respectively. Let Xhd and W h

d in �d contain at least polynomials ofdegree rd and ld , respectively. In all cases under consideration, either ld = rd orld = rd − 1. Let �H

sd , �Hd , and �H

s contain at least polynomials of degree rsd , rdd ,and rss , respectively. In the analysis we will make use of the following well knownapproximation properties of the mixed interpolant Rh [18,34]: for all v ∈ Ht (�)n ,t ≥ 1, there exists a constant C that depends only on rd , ld , t , and the shape regularityof T h

i , such that for all T in T hi

‖v − Rh(v)‖L2(T ) ≤ Chr |v|Hr (T ), 1 ≤ r ≤ min(rd + 1, t), (5.1)

‖div(v − Rh(v))‖L2(T ) ≤ Chr |div v|Hr (T ), 0 ≤ r ≤ min(ld + 1, t − 1), (5.2)

‖(v − Rh(v)|�d,i ) · n‖L2(T ′) ≤ Chr |v · n|Hr (T ′), 0 ≤ r ≤ min

(rd + 1, t − 1

2

),

(5.3)

where T ′ denotes an arbitrary element of the trace T hi,�i j

on �i j . We begin with the

following approximation result for the operator �h defined in Lemma 3.6.

Lemma 5.1 Under the assumptions of Lemma 3.6, the operator �h ∈L(H10 (�)n; V h)

satisfies for all v ∈ (Ht (�) ∩ H10 (�)

)n, t ≥ 1,

|||v − �h(v)|||Xs≤ Chr |v|Hr+1(�), 0 ≤ r ≤ min(rs, t − 1), (5.4)

|||v − �h(v)|||Xd≤ C

(hr‖v‖

Hr+ 12 (�)

+ hq‖div v‖Hq (�) + hs‖v‖Hs+1(�)

),

1

2≤r ≤ min

(rd + 1, t − 1

2

), 0≤q ≤ min(ld + 1, t−1), 0 ≤ s ≤ min(rs, t−1).

(5.5)

Proof To show (5.4), recall that �h(v)|�s = �hs (v), �h

s (v) = �hs (v) + ch

s (v), wherech

s (v) is defined in (3.29), and �hs is a correction of the Scott-Zhang operator con-

structed in Sect. 4. The triangle inequality and (3.30) imply that

123


‖v − �hs (v)‖H1(�s,i )

≤ C‖v − �hs (v)‖H1(�s,i )

.

Recall that �hs (v) is constructed by correcting the Scott-Zhang interpolant Sh or

Sh with chi,�i j

(v) and chj,�i j

(v). Since ‖v − Sh(v)‖H1(�s,i )is bounded optimally in

(4.16), it remains to bound ‖chi,�i j

(v)‖H1(�s,i )and ‖ch

j,�i j(v)‖H1(�s, j )

. The bound on

‖chi,�i j

(v)‖H1(�s,i )follows from a simple modification of the argument in the proof of

Lemma 4.1. More precisely, by using (4.15) with 0 ≤ r ≤ min(rs, t − 1), (4.20)can be modified as

∣∣∣∣∣∣∣

∫

�k

(v − Sh(v)|�s,i

)

∣∣∣∣∣∣∣≤ Chr

T |�k | ρ− 12

k |v|Hr+1(Dk ).

Propagating the above bound through the proof gives

‖chi,�i j

(v)‖L2(�s,i )+h|ch

i,�i j(v)|H1(�s,i )

≤Chr+1|v|Hr+1(�s,i ), 0≤r ≤min(rs, t − 1).

We proceed with the bound on ‖chj,�i j

(v)‖H1(�s, j ). Considering first the 2-D case

presented in Sect. 4.3, inequality (4.34) in the proof of Lemma 4.3 can be modified,using the approximation property (4.16) of Sh , as

|(Sh(v) − v)|�s,k |H

12 (�i j )

≤ Chr |v|Hr+1(�s,k), 0 ≤ r ≤ min(rs, t − 1),

and (4.38) and (4.39) can be modified, using the approximation property (4.15) of Sh ,as

xn+1∫

xn

1

x

∣∣∣[Sh(v)]∣∣∣2 ≤ Ch2r |v|2

Hr+1(�n,i ∪�n, j ), 0 ≤ r ≤ min(rs, t − 1),

implying

‖chj,�i j

(v)‖H1(�s, j )≤ Chr |v|Hr+1(�s,i ∪�s, j )

, 0 ≤ r ≤ min(rs, t − 1).

We next consider the 3-D case presented in Sect. 7.2. Modifying the proof ofLemma 7.4 in a similar way gives

‖chj,�i j

(v)‖L2(�s, j )+ h|ch

j,�i j(v)|H1(�s, j )

≤ Chr+1|v|Hr+1(�s,i ∪�s, j ), 0 ≤ r ≤ min(rs , t − 1).

A combination of the above inequalities gives (5.4).We continue with the proof of (5.5). Recall that �h(v)|�d = �h

d(v), where�h

d(v) = Rh(v) + chj,�i j

(v) is a corrected mixed interpolant with a correction sat-

isfying (4.8). Since ‖v − Rh(v)‖H(div;�d,i ) is bounded optimally in (5.1)–(5.2), it

123


remains to bound chj,�i j

(v) constructed in Sect. 4.4. By construction div chj,�i j

(v) = 0,

so we only need to control ‖chj,�i j

(v)‖L2(�d, j ). Consider first �i j ∈ �dd . Using (5.3),

bound (4.49) gives

‖chj,�i j

(v)‖L2(�d, j )≤ C A

12j hr |v|Hr (�i j ) ≤ Chr‖v‖

Hr+ 12 (�d,i ∪�d, j )

,

0 < r ≤ min

(rd +1, t − 1

2

),

where we have used the trace inequality [38]

|ϕ|Hr (�i j ) ≤ A− 1

2i ‖ϕ‖

Hr+ 12 (�d,i )

, r > 0. (5.6)

For �i j ∈ �sd , we modify the argument in Corollary 4.3 to use the full approximationproperties (4.16) of Sh or Sh to obtain

‖chj,�i j

(v)‖L2(�d, j )≤ C

(hr‖v‖

Hr+ 12 (�s,i ∪�d, j )

+ hs‖v‖Hs+1(�s,i ∪�d, j )

),

0 < r ≤ min

(rd + 1, t − 1

2

), 0 ≤ s ≤ min(rs, t − 1).

A combination of the above inequalities completes the proof of (5.5).

Next, we need to approximate the functions for the pressure. For any q ∈ L2(�i ), letPhq be its L2(�i )-projection onto W h

i := W h |�i ,

(q − Phq, wh) = 0, ∀wh ∈ W hi ,

satisfying the approximation property

‖q − Phq‖0,�i ≤ C hr |q|Hr (�i ), 0 ≤ r ≤ ri + 1, (5.7)

where ri is the polynomial degree in the space W hi : ri = rs − 1 in �s and ri = ld

in �d .

5.1 Error estimates

From the error equation:

∀vh ∈ Zh,∀qh ∈ Wh, a(u − uh, vh) = −(b(vh, p − qh) + bsd(vh, λsd)

+bd(vh, λd) + bs(vh,λs)), (5.8)

123


and Lemmas 3.3 and 3.4, we immediately derive

|||u − uh |||X ≤ C

(inf

vh∈V h(qd )|||u − vh |||X + inf

wh∈W h‖p − wh‖W

)+ CRh

u, (5.9)

where V h(qd) = {v ∈ V h; ∀w ∈ W h, b(v, w) = ∫�d

w qd}, and

Rhu = sup

vh∈Zh

1

|||vh |||X

∣∣∣bsd(vh, λsd) + bd(vh, λd) + bs(vh,λs)

∣∣∣ (5.10)

is the consistency error at the interfaces. Similarly a simple variant of (5.8) withvh ∈ V h and Theorem 3.1 yield

‖p − ph‖W ≤ C

(|||u − uh |||X + inf

wh∈W h‖p − wh‖W

)+ CRh

p, (5.11)

where Rhp is given by (5.10) with Zh replaced by V h . As the bound on the approxi-

mation error of W h follows from (5.7) and Lemma 5.1 estimates the approximationerror of V h(qd), it suffices to bound Rh

p. Note that by virtue of (3.11), we have for all

μHs ∈ �H

s , μHd ∈ �H

d , μHsd ∈ �H

sd ,

Rhp = sup

vh∈V h

1

|||vh |||X

∣∣∣bsd(vh, λsd −μH

sd)+bd(vh, λd −μHd )+bs(v

h,λs −μHs )

∣∣∣ .

(5.12)

The bound for bs(vh,λs − μH

s ) is straightforward because vh is measured in H1 ineach �s,i , and therefore its trace can be considered on each �i, j . But deriving anoptimal bound for the other terms is more intricate because vh is measured in H(div)

in each �d,i and this only controls the trace of the normal component in H− 12 (∂�d,i ).

Consequently, λsd − μHsd and λd − μH

d must belong to H12 (∂�d,i ). In 2-D, this is

easily achieved by interpolating λsd and λd with a variant of the Scott and Zhanginterpolant that is continuous at the vertices of the subdomains; a similar idea wasused in Sect. 4.3. But in 3-D, this construction requires an additional assumption onthe mortar grids in the Darcy region.

Hypothesis 5.1 For each �d,i in 3-D, the mortar grids T Hi j on all �i j ⊂ ∂�d,i\∂�

are chosen such that their traces on the boundaries of �i j coincide.

This assumption implies conformity of mortar grids along interface boundaries ineach connected region in �d and it allows us to consider the space �

H,cd , which is the

subset of continuous functions in �Hd ∪�H

sd . On �dd ∪�sd , let I H be the Scott–Zhang

interpolant in �H,cd . We assume that its definition depends only on function values on

�dd ∪�sd . On �ss , let I H be the L2(�i j )-orthogonal projection operator in �Hs . This

operator has the following approximation properties, assuming sufficient smoothnessof μ and μ (the index � stands for dd or sd):

123


∀�i j ∈ ��,∀τ ∈�i j , ‖μ−IH (μ)‖Ht (τ ) ≤C Hr−t |μ|Hr (�τ ), 0≤ t ≤1, t ≤r ≤r�+1,

(5.13)

∀�i j ∈ �ss,∀τ ∈ �i j , ‖μ − IH (μ)‖L2(τ ) ≤ C Hr |μ|Hr (�τ ), 0 ≤ r ≤ rss + 1,

(5.14)

where �τ is the macroelement used in defining I H (μ) restricted to τ . The union ofall �τ for τ in ∂�i may slightly overlap interfaces that are not part of ∂�i ; we denotethe overlap by O.

Lemma 5.2 There exists a constant C independent of h, H, and the diameters of thesubdomains such that, for all vh ∈ Xh:

|bs(vh,λs − I H (λs))| ≤ C Hs

Ms∑

i=1

A− 1

2i ‖vh‖H1(�s,i )

|λs |Hs (∂�s,i ∩�ss∪O),

0 ≤ s ≤ rss + 1, (5.15)

provided λs is sufficiently smooth.

Proof We have

bs(vh,λs − I H (λs)) =

∑

�i j ∈�ss

∫

�i j

[vh] · (λs − I H (λs))

≤Ms∑

i=1

‖vh‖L2(∂�s,i ∩�ss )‖λs − I H (λs)‖L2(∂�s,i ∩�ss )

.

Then (5.15) follows readily from (5.14) and the trace inequality (4.51).

Lemma 5.3 Under Hypothesis 5.1, there exists a constant C independent of h, H,and the diameters of the subdomains such that, for all vh ∈ Xh:

|bsd(vh, λsd − I H (λsd)) + bd(vh, λd − I H (λd))|

≤C

⎛

⎝Md∑

i=1

‖vh‖H(div;�d,i )

(Hq− 1

2 |λ|Hq (∂�d,i ∩�dd∪O)+Hr− 12 |λ|Hr (∂�d,i ∩�sd∪O)

)

+Md∑

i=1

∑

j

A− 1

2j ‖vh‖H1(�s, j )

Hr |λ|Hr (∂�d,i ∩�sd∪O)

⎞

⎠ ,

1

2≤q ≤ rdd +1,

1

2≤r ≤rsd + 1, (5.16)

provided λsd and λd are sufficiently smooth, and where the sum over j runs over all�s, j adjacent to �d,i .

Proof In the following λ denotes either λd or λsd depending on the type of interface.We can write

123


bsd(vh, λ − I H (λ)) + bd(vh, λ − I H (λ))

=∑

�i j ∈�sd∪�dd

∫

�i j

[vh · n](λ − I H (λ))

=Md∑

i=1

∫

∂�d,i

vh · ni (λ − I H (λ)) +∑

�i j ∈�sd

∫

�i j

vh · ns(λ − I H (λ)),

where we have used that vh · n = 0 on ∂�d,i ∩ ∂� and λ − I H (λ) has been extended

continuously from H12 (∂�d,i\∂�) to H

12 (∂�d,i ). The argument of Lemma 5.2 can

be used to bound the second sum above by the last sum in (5.16). Thus it is left tobound the first sum.

Consider first one subdomain �d,i that is not adjacent to �sd . Let us switch to thereference domain �i and let E denote the extension operator defined in (4.29) relative

to �i . For any f in H12 (∂�d,i ), define E( f ) by: E( f ) = E( f ◦ Fi ); therefore E

maps H12 (∂�d,i ) into H1(�d,i ). Thus, by Green’s formula

∫

∂�d,i

vh · ni (λ − I H (λ)) =∫

∂�d,i

vh · ni E(λ − I H (λ))

=∫

�d,i

div vh E(λ − I H (λ)) +∫

�d,i

vh · ∇ E(λ − I H (λ)).

By reverting to the reference domain and observing that I H is invariant under the rigidbody motion Fi , this reads

∣∣∣∣∣∣∣

∫

∂�d,i

vh · ni (λ − I H (λ))

∣∣∣∣∣∣∣≤ An−1

i ‖v‖H( ˆdiv;�i )‖E(λ − I(λ))‖H1(�i )

.

On one hand, the bound in (4.29) gives

‖E(λ − I(λ))‖H1(�i )≤ C‖λ − I(λ))‖

H12 (∂�i )

≤ C‖λ − I(λ))‖H

12 (∂�i \∂�)

≤ C( ∑

�⊂∂�i \∂�

‖λ − I(λ))‖2

H12 (�)

) 12,

with constants here and below independent of h, H , and the diameters of �d,i and�i j . By space interpolation between L2(�) and H1(�), the estimates (5.13) summedon � with t = 0 and t = 1 readily imply that

‖λ − I(λ))‖H

12 (�)

≤ C Hr− 1

2i |λ|Hr (�

�),

1

2≤ r ≤ rdd + 1, (5.17)

123


where the mesh size Hi on ∂�i is related to the mesh size Hi on ∂�i by

Hi = Ai Hi .

In the case when r is not an integer, since (5.17) is stated in the reference region �,we choose the fractional seminorm in the right-hand side to be defined by (1.8) andincorporate the equivalence constant into C . The motivation for this choice is that theintrinsic norm is easily transformed by Fi . Therefore

‖λ − I(λ))‖H

12 (∂�i )

≤ C Hr− 1

2i |λ|Hr (∂�i \∂� ∪O)

≤ C Hr− 1

2i A

r− n−12

i |λ|Hr (∂�d,i \∂� ∪O).

On the other hand,

‖v‖H( ˆdiv;�i )= A

− n2

i

(A2

i ‖div vh‖2L2(�d,i )

+ ‖vh‖2L2(�d,i )

) 12.

By collecting these inequalities, we derive

∣∣∣∣∣∣∣

∫

∂�d,i

vh · ni (λ − I H (λ))

∣∣∣∣∣∣∣≤ C

(Ai Hi

)r− 12 ‖vh‖H(div;�d,i )|λ|Hr (∂�d,i \∂� ∪O),

1

2≤ r ≤ rdd + 1.

By applying (5.17) to a portion of �sd , using (5.13) the same argument handles thecase when �d,i is adjacent to �sd . Then (5.16) follows easily from these estimates.

Remark 5.1 We can skip Hypothesis 5.1, work locally on each �i j , and use the L2

norm for both factors vh · ni and λ − I H (λ). But this gives rise to a factor of the form(Hi/h

) 12 in (5.16), that is clearly not optimal when h is very small compared to Hi .

Let Rh denote either Rhu or Rh

p. The next lemma estimates Rh .

Lemma 5.4 Under Hypothesis 5.1, the consistency error Rh satisfies

Rh ≤ C(

A− 12 Hr− 1

2 ‖pd‖Hr+ 1

2 (�d )+ A− 1

2 Hq− 12 ‖pd‖

Hq+ 12 (�d )

+A−1 Hs(‖us‖Hs+ 3

2 (�s )+ ‖ps‖

Hs+ 12 (�s)

)),

1

2≤ q ≤ rdd + 1,

1

2≤ r ≤ rsd + 1, 0 < s ≤ rss + 1, (5.18)

where A = min1≤i≤M Ai .

123


Proof By combining (5.15) and (5.16), we easily derive

Rh ≤ C

⎛

⎜⎝Hq− 1

2

⎛

⎝Md∑

i=1

|λ|2Hq (∂�d,i ∩�dd∪O)

⎞

⎠

12

+ Hr− 12

⎛

⎝Md∑

i=1

|λ|2Hr (∂�d,i ∩�sd∪O)

⎞

⎠

12

+Hs

( Ms∑

i=1

A−1i |λs |2Hs (∂�s,i ∩�ss∪O)

) 12

+Hr

( Ms∑

i=1

A−1i |λ|2Hr (∂�s,i ∩�sd∪O)

) 12

⎞

⎟⎠ ,

1

2≤ q ≤ rdd + 1,

1

2≤ r ≤ rsd + 1, 0 ≤ s ≤ rss + 1.

In view of the representation (2.47), local trace theorems yield:

|λsd |Hr (∂�d,i ∩�sd ) ≤ C A− 1

2i ‖pd‖

Hr+ 12 (�d,i )

,

|λd |Hq (∂�d,i ∩�dd ) ≤ C A− 1

2i ‖pd‖

Hq+ 12 (�d,i )

,

|λs |Hs (∂�s,i ∩�ss ) ≤ C A− 1

2i

(‖us‖Hs+ 3

2 (�s,i )+ ‖ps‖

Hs+ 12 (�s,i )

).

Then (5.18) follows from these bounds, that are easily extended to include O, and thefact that A−1

i < H−1.

Now we use the abstract error bounds (5.9) and (5.11), the approximation results(5.4), (5.5), and (5.7), and the consistency error bound (5.18) to obtain the followingconvergence result.

Theorem 5.1 We have

|||u − uh |||X + ‖p − ph‖W

≤ C(hr1(‖u‖Hr1+1(�) + ‖p‖Hr1 (�)) + hr2‖u‖

Hr2+ 12 (�)

+ hr3(‖div u‖Hr3 (�) + ‖p‖Hr3 (�))

+ A−1 Hr4(‖us‖Hr4+ 3

2 (�s )+ ‖ps‖

Hr4+ 12 (�s)

)

+ A− 12 Hr5− 1

2 ‖pd‖Hr5+ 1

2 (�d )+ A− 1

2 Hr6− 12 ‖pd‖

Hr6+ 12 (�d )

),

0 ≤ r1 ≤ rs,1

2≤ r2 ≤ rd + 1, 0 ≤ r3 ≤ ld + 1,

0 < r4 ≤ rss + 1,1

2≤ r5 ≤ rdd + 1,

1

2≤ r6 ≤ rsd + 1.

123


Remark 5.2 In the above estimate, the fine scale subdomain approximation error termsare of optimal order with constants independent of the size of the subdomains A. Theconstants of the coarse scale mortar consistency error terms deteriorate with decreasein A, since in that case the number of interfaces grows. Nevertheless, higher ordermortar polynomials can be employed to balance the error terms, giving optimal finescale convergence.

6 Numerical tests

In this section we validate our analysis by carrying out some numerical experiments.In all tests the computational domain is taken to be � = �s ∪ �d , where �s =(0, 1) × ( 1

2 , 1) and �d = (0, 1) × (0, 12 ). For simplicity we set

σ (us, ps) = −ps I + νs∇us

in the Stokes equation in �s , and

K = K I

in the Darcy equation in �d , where K is a positive constant.To test for convergence we construct the following analytical solution satisfying

the flow equations in �s and �d along with the conditions on the interface �sd :

us =[

(2 − x)(1.5 − y)(y − ξ)

− y3

3 + y2

2 (ξ + 1.5) − 1.5ξ y − 0.5 + sin(ωx)

]

,

ud =[

ω cos(ωx)yχ(y + 0.5) + sin(ωx)

],

ps = − sin(ωx) + χ

2K+ νs(0.5 − ξ) + cos(πy),

pd = − χ

K

(y + 0.5)2

2− sin(ωx)y

K,

where

νs =0.1, K =1, α=0.5, G =√

νs K

α, ξ = 1−G

2(1 + G), χ =−30ξ−17

48, and ω=6.0.

The right-hand sides f s , f d , and qd for the Stokes–Darcy flow system are obtainedby substituting the analytical solution into (2.6), (2.3), and (2.4), respectively. Theboundary conditions are as follows: for the Stokes region, the velocity us is specifiedon the left and top boundaries, and the normal and tangential stresses (σ ns) · ns

and (σ ns) · τ s are specified on the right boundary; for the Darcy region, the normalvelocity ud · nd is specified on the left boundary and the pressure pd is specifiedon the bottom and right boundaries. Each region �s and �d is divided into two

123


Fig. 2 Non-matchingsubdomain meshes8 × 12–12 × 16

X

Y

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

subdomains, giving a total of four subdomains. The subdomain grids do not matchacross the interfaces. The Stokes subdomains are discretized by the Taylor–Hoodtriangular finite elements with quadratic velocities and linear pressures (rs = 2). TheDarcy subdomains are discretized by the lowest order Raviart–Thomas rectangularfinite elements (rd = ld = 0). We use discontinuous piecewise linear mortars on allinterfaces (rss = rdd = rsd = 1). To test convergence, we solve the problem on asequence of grid refinements. On the coarsest level, the subdomain grids are 3 × 4 inthe lower left and upper right subdomains and 2 × 3 in the other two subdomains. Wetest two cases, H = 2h and H = √

h. In both cases the coarsest mortar grids have asingle element per interface. In the first case the mortar grids are refined by two eachtime the subdomain grids are refined by two. In the second case the mortar grids arerefined by two each time the subdomain grids are refined by four. The non-matchinggrids on the middle level of refinement are shown in Fig. 2. The computed solutionand the numerical error on this grid level with mortar meshes H = 2h are displayed inFig. 3. The numerical errors and convergence rates on all refinement levels are reportedin Tables 1, 2, 3, 4, 5, and 6. We report separately the errors in �s and �d in theirrespective norms. In the Darcy region we also report the discrete L2 errors ‖ · ‖M,�d

for the pressure and the velocity, computed via the use of the midpoint quadrature ruleon each element.

In the case H = 2h we observe second order of convergence for ‖us − uhs ‖H1(�s )

and ‖ps − phs ‖L2(�s )

, as well as first order convergence for ‖ud − uhd‖H(div;�d ) and

‖pd − phd‖L2(�d ). These rates are consistent with the error terms O(hrs ) = O(h2) and

O(hrd+1) = O(hld+1) = O(h) that appear in Theorem 5.1. Note that in the case H =2h the interface consistency error terms are O(hrss+1) = O(h2) and O(hrdd+ 1

2 ) =O(hrsd+ 1

2 ) = O(h32 ). While the error terms in Theorem 5.1 depend on the global

solution norms, the results indicate that the lower convergence rates in �d and on �sd

do not affect the second order convergence in �s . This suggests that it may be possible

123


X

Y

0

0.2

0.4

0.6

0.8

1

P

0.60.40.20

-0.2-0.4-0.6-0.8-1-1.2

3

X

Y

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

P-error

0.0140.0120.010.0080.0060.0040.0020

-0.002-0.004-0.006-0.008-0.01-0.012-0.014-0.016-0.018-0.02-0.022

0.1

Fig. 3 Test 1: computed solution (left) and error (right) on subdomain meshes 8 × 12–12 × 16 and mortarmeshes H = 2h

Table 1 Test 1: H = 2h

Mesh ‖us − uhs ‖H1(�s )

Rate ‖ps − phs ‖L2(�s )

Rate

2×3 3×4 3.93e−01 2.64e−02

4×6 6×8 8.95e−02 2.13 6.37e−03 2.05

8×12 12×16 2.10e−02 2.09 1.53e−03 2.06

16×24 24×32 5.08e−03 2.05 3.75e−04 2.03

32×48 48×64 1.25e−03 2.02 9.29e−05 2.01

Numerical errors and convergence rates in �s


Mesh ‖ud − uhd‖H(div;�d ) Rate ‖pd − ph

d ‖L2(�d ) Rate

2×3 3×4 3.51e+00 1.01e−01

4×6 6×8 1.79e+00 0.97 5.05e−02 1.00

8×12 12×16 9.00e−01 0.99 2.52e−02 1.00

16×24 24×32 4.50e−01 1.00 1.26e−02 1.00

32×48 48×64 2.20e−01 1.00 6.28e−03 1.00

Numerical errors and convergence rates in �d

to establish interior convergence error estimates. Such estimates were derived in [53]for mortar discretizations for Darcy flow. We also observe superconvergence at the cellcenters in the Darcy region, as indicated by the O(h2) convergence of ‖pd − ph

d‖M,�d

and the O(h32 ) convergence of ‖ud − uh

d‖M,�d . Superconvergence for the pressure onunstructured grids and for the velocity on logically rectangular grids in mixed finiteelement discretizations for Darcy flow has been extensively studied in the literature,see, e.g. [5] and the references therein. The reduced superconvergence order for the

Darcy velocity is due to the O(h32 ) mortar error term, as shown in [5].

123



Mesh ‖ud − uhd‖M,�d Rate ‖pd − ph

d ‖M,�d Rate

2×3 3×4 2.60e−01 1.65e−02

4×6 6×8 6.71e−02 1.95 4.13e−03 2.00

8×12 12×16 2.09e−02 1.68 1.01e−03 2.03

16×24 24×32 6.85e−03 1.61 2.50e−04 2.01

32×48 48×64 2.32e−03 1.56 6.22e−05 2.01

Numerical errors and convergence rates in �d using the discrete ‖ · ‖M,�d norm

Table 4 Test 2: H = √h

Mesh ‖us − uhs ‖H1(�s )

Rate ‖ps − phs ‖L2(�s )

Rate

2×3 3×4 3.93e−01 2.64e−02

8×12 12×16 5.59e−02 1.41 3.51e−03 1.46

32×48 48×64 1.03e−02 1.22 6.02e−04 1.27

Numerical errors and convergence rates in �s


Mesh ‖ud − uhd‖H(div;�d ) Rate ‖pd − ph

d ‖L2(�d ) Rate

2×3 3×4 3.51e+00 1.01e−01

8×12 12×16 9.10e−01 0.97 2.53e−02 1.00

32×48 48×64 2.30e−01 0.99 6.32e−03 1.00

Numerical errors and convergence rates in �d


Mesh ‖ud − uhd‖M,�d Rate ‖pd − ph

d ‖M,�d Rate

2×3 3×4 2.60e−01 1.65e−02

8×12 12×16 1.10e−01 0.62 3.65e−03 1.09

32×48 48×64 5.01e−02 0.57 7.57e−04 1.13

Numerical errors and convergence rates in �d using the discrete ‖ · ‖M,�d norm

In the case H = √h, we observe approximately O(h) convergence for all error

norms. Note that in this case the interface consistency error terms are O(h(rss+1)/2) =O(h) and O(h(rdd+ 1

2 )/2) = O(h(rsd+ 12 )/2) = O(h

34 ), so their effect on the conver-

gence in the Stokes and Darcy regions is more significant. In this multiscale case, onemay utilize higher order mortars to recover optimal fine scale subdomain convergence,see [6] for the Darcy case.

123


7 Appendix

This Appendix is devoted to specific examples of construction of chj,�i j

(v) in �s .

7.1 The 2-D case

Recall that the construction of the correction chj,�i j

(v) in �s in 2-D required an operator

πhs satisfying (4.23). In particular, given � in H

12

00(�)2, one needs to construct π j (�)

in X j , unique solution of (4.24):

∀μ ∈ �,

∫

�

π j (�) · μ =∫

�

� · μ,

and satisfying (4.25).Here we present two simple examples of how to construct such an operator when

�i j is a straight line segment and the traces of the discrete spaces on �i j are piecewiseP1 finite elements, Without loss of generality, we can suppose that � is the segment[0, 1], that the nodes of T j are 0 = x0 < x1 < · · · < xN < xN+1 = 1, and we sethn = xn+1 − xn , 0 ≤ n ≤ N . We choose each component of the functions of X j

piecewise P1 in the subintervals [xn, xn+1], with degrees of freedom at the nodes xn ,1 ≤ n ≤ N .

7.1.1 First example

As a first example, we choose each component of the mortar functions of � alsopiecewise P1 in the subintervals [xn, xn+1], with degrees of freedom at the nodes xn ,n = 0, 2 ≤ n ≤ N − 1, and n = N + 1. The nodes x1 and xN are deleted so thatX j and � have the same dimension 2N ; thus the matrix of the linear system (4.24)is square. In order to express these functions in terms of a basis, it is convenient tomodify slightly the standard Lagrange basis functions for the velocity. More precisely,we define

ϕ0(x)|[x0,x2] = 1

h0 + h1(x2 − x), ϕN+1(x)|[xN−1,xN+1] = 1

hN + hN−1(x − xN−1),

ϕ2(x)|[x0,x2] = 1

h0 + h1(x − x0), ϕ2(x)|[x2,x3] = 1

h2(x3 − x),

ϕN−1(x)|[xN−2,xN−1] =1

hN−1(x− xN−2), (7.1)

ϕN−1(x)|[xN−1,xN+1] = 1

hN−1 + hN(xN+1 − x),

ϕn(x)|[xn−1,xn ] = 1

hn−1(x − xn−1), ϕn(x)|[xn ,xn+1] = 1

hn(xn+1 − x),

n = 1, 3 ≤ n ≤ N − 2, n = N ,

123


extended by zero elsewhere. We take

X j ={

v∈ H10 (0, 1)2; v(x)=

N∑

n=1

vn ϕn(x)

}

, (7.2)

�={

μ∈ H1(0, 1)2; μ(x)= μ(x0)ϕ0(x)+N−1∑

n=2

μ(xn)ϕn(x)+μ(xN+1)ϕN+1(x)

}

.

(7.3)

On one hand, the set {ϕ0, ϕn, 2 ≤ n ≤ N − 1, ϕN+1} is indeed a Lagrange basis for�. On the other hand, the set {ϕn, 1 ≤ n ≤ N } is a basis but not a Lagrange basis forX j ; however, it is easy to check that

v1 = v(x1) − v(x2)ϕ2(x1), vn = v(xn), 2 ≤ n ≤ N − 1,

vN = v(xN ) − v(xN−1)ϕN−1(xN ). (7.4)

The system (4.24) can be decoupled into two independent systems inRN , one for eachcomponent, and each system has the form Mα = b, where α ∈ RN is the unknown,the non zero coefficients of M per row are

M1,1 = 1

6(h0 + 2h1), M1,2 = 1

6(h0 + h1),

M2,1 = 1

6(2h0 + h1), M2,2 = 1

3(h0 + h1 + h2), M2,3 = h2

6,

Mn,n−1 = hn−1

6, Mn,n = 1

3(hn−1 + hn), Mn,n+1 = hn

6, 3 ≤ n ≤ N − 2,

MN−1,N−2 = hN−2

6, MN−1,N−1 = 1

3(hN−2 + hN−1 + hN ),

MN−1,N = 1

6(hN−1 + 2hN ), MN−1,N = 1

6(2hN−1 + hN ),

MN ,N = 1

6(hN−1 + hN ).

Denoting by � a generic component of �, the components of b ∈ RN are

b1 =x2∫

x0

�ϕ0, b2 =x3∫

x0

�ϕ2, bn =xn+1∫

xn−1

�ϕn, 3 ≤ n ≤ N − 2,

bN−1 =xN+1∫

xN−2

�ϕN−1, bN =xN+1∫

xN−1

�ϕN+1.

(7.5)

The matrix M is tridiagonal but not symmetric, the coefficients of its three diagonalsare all strictly positive, and it is strictly diagonally dominant, hence invertible, but

123


this is not sufficient to obtain a sharp estimate for its inverse. To this end, we usethe approach of Crouzeix and Thomée [23]. More precisely, let D be the principaldiagonal of M and factor M into M = D(I + K ) where K is a tridiagonal matrixwith principal diagonal zero. Denote the diagonal terms of D by dn > 0 and note that

D12 is well-defined. The next lemma relates π j (�) and α. Recall that | · | denotes the

Euclidean vector norm.

Lemma 7.1 For each component � of � we have

‖π j (�)‖L2(�)≤ 2|D

12 α|. (7.6)

Proof Considering the support of the basis functions ϕn , we have

‖π j (�)‖2L2(�)

=x2∫

x0

(α1ϕ1 + α2ϕ2)2 +

N−2∑

n=2

xn+1∫

xn

(αn ϕn + αn+1ϕn+1)2

+xN+1∫

xN−1

(αN−1ϕN−1 + αN+1ϕN+1)2.

By substituting the expression of these integrals and rearranging terms, we derive

‖π j (�)‖2L2(�)

≤ 2

3

(α2

1(h0 + h1) + α22(h0 + h1 + h2) +

N−2∑

n=3

α2n(hn−1 + hn)

+α2N−1(hN−2 + hN−1 + hN ) + α2

N (hN−1 + hN )).

In view of the diagonal terms dn of D, this can be written

‖π j (�)‖2L2(�)

≤ 2

(

α21

h0 + h1

3+

N−1∑

n=2

α2ndn + α2

NhN−1 + hN

3

)

.

But

h0 + h1

3= 2d1

h0 + h1

h0 + 2h1< 2d1, and similarly

hN−1 + hN

3< 2dN .

Hence

‖π j (�)‖L2(�)≤ 2

(N∑

n=1

α2ndn

) 12

, (7.7)

whence (7.6).

123


This lemma shows that an L2 bound for π j (�) relies on a bound for |D12 α|. But D

12 α

has also the following expression

D12 α = (I + D

12 K D− 1

2)−1(D− 1

2 b).

Therefore Lemma 7.1 implies that

‖π j (�)‖L2(�)≤ 2‖(I + D

12 K D− 1

2)−1‖2|D− 1

2 b|, (7.8)

where ‖ · ‖2 denotes the matrix norm subordinated by the Euclidean norm, and moregenerally ‖·‖p is the matrix norm subordinated by the l p vector norm. The next lemma

gives a bound for |D− 12 b|.

Lemma 7.2 We have

|D− 12 b| ≤ √

3‖�‖L2(�). (7.9)

Proof By integrating the basis functions ϕn , we derive

|D− 12 b|2 =

N∑

n=1

d−1n b2

n

≤ 1

3

(h0 + h1

d1‖�‖2

L2(x0,x2)+ h0 + h1 + h2

d2‖�‖2

L2(x0,x3)

+N−2∑

n=3

hn−1+hn

dn‖�‖2

L2(xn−1,xn+1)+ hN−2+hN−1+hN

dN−1‖�‖2

L2(xN−2,xN+1)

+hN−1+hN

dN‖�‖2

L2(xN−1,xN+1)

)

. (7.10)

Then proceeding as in Lemma 7.1, we obtain

|D− 12 b|2 ≤ ‖�‖2

L2(x0,x1)+ ‖�‖2

L2(x0,x2)+

N∑

n=1

‖�‖2L2(xn−1,xn+1)

+ ‖�‖2L2(xN ,xN+1)

+‖�‖2L2(xN−1,xN+1)

,

whence (7.9).

It remains to evaluate ‖(I + D12 K D− 1

2)−1‖2. Following Crouzeix and Thomée, we

introduce the following constraint on the mesh length hn :

123


Hypothesis 7.1 There exist two constants c0 > 0 and γ > 0 independent of N suchthat

∀n, m, 0 ≤ n, m ≤ N ,hn

hm≤ c0γ

|n−m|. (7.11)

This condition holds for quasi-uniform meshes, but it is also satisfied by much moregeneral meshes.

Proposition 7.1 If Hypothesis 7.1 holds with suitable constants c0 and γ , then thereexists a constant C independent of N and j such that

‖(I + D12 K D− 1

2)−1‖2 ≤ C. (7.12)

Hence with the same constant C,

∀� ∈ L2(�)2, ‖π j (�)‖L2(�)≤ 2

√3C‖�‖L2(�)

. (7.13)

Proof The proof follows the ideas of [23] but is more complex because the functions ofX j vanish at the end points of � whereas those of � do not. Let us prove convergenceof the series

∞∑

r=1

‖D12 K r D− 1

2 ‖2;

this will yield

‖(I + D12 K D− 1

2)−1‖2 ≤ 1 +

∞∑

r=1

‖D12 K r D− 1

2 ‖2. (7.14)

As K has three diagonals, then K r has 2r + 1 diagonals. In addition, since the coef-ficients of these diagonals are non negative, this implies that

‖D12 K r D− 1

2 ‖1 ≤ (2r + 1)‖D12 K r D− 1

2 ‖∞,

and by interpolation,

‖D12 K r D− 1

2 ‖2 ≤ (2r + 1)12 ‖D

12 K r D− 1

2 ‖∞.

Therefore

‖D12 K r D− 1

2 ‖2 ≤ sup|n−m|≤2r

( dn

dm

) 12(2r + 1)

12 ‖K‖r∞. (7.15)

123


Thus a sufficient condition for the series convergence is:

sup|n−m|≤2r

( dn

dm

) 12 ≤ c

( δ

‖K‖∞

)r, (7.16)

where c > 0 and 0 < δ < 1 are two constants independent of r . First, assumingHypothesis 7.1, a simple but tedious computation gives for 0 < γ < 2

sup|n−m|≤2r

dn

dm≤ 2c0(1 + γ + γ 2)γ 2r . (7.17)

Hence the series converges if

γ < Min

(2,

1

‖K‖∞

).

It is easy to check that

‖K‖∞ = Max

(h0 + h1

h0 + 2h1,

hN + hN−1

hN + 2hN−1,

1

2

(

1 + h0

h0 + h1 + h2

)

,

1

2

(

1 + hN

hN−2 + hN−1 + hN

))

.

To simplify the notation, set θ = c0γ . Owing to Hypothesis 7.1, for θ ≥ 1, we have

‖K‖∞ ≤ 1

2

(1 + Max

( θ

2 + θ,

θ2

1 + θ + θ2

)) = 1

2

(1 + θ2

1 + θ + θ2

). (7.18)

Take for instance θ = 32 ; then (7.18) yields ‖K‖∞ < 3

4 . Therefore the series convergesfor γ = 4

3 and in turn c0 = 98 . Then (7.13) is an immediate consequence of Lemmas

7.1 and 7.2.

Now we estimate π j (�) in H1(�)2 for � ∈ H10 (�)2. First, we have the analogue of

Lemma 7.1.

Lemma 7.3 We keep the notation of Lemma 7.1. Under Hypothesis 7.1 with γ > 1,we have

|π j (�)|H1(�)≤√

2

3

(c0γ (1 + γ ) + 2 + c0γ

2

1 + γ

) 12

|D− 12 α|. (7.19)

Proof Let us denote the derivation on � with a prime. Then

π j (�)′ =

N∑

n=1

αn ϕ′n,

123


and a straightforward computation gives

‖π j (�)′‖2

L2(�)≤ 2

(

α21

(1

h0+ 1

h1

)+α2

2

(1

h0 + h1+ 1

h2

)+

N−2∑

n=3

α2n

(1

hn−1+ 1

hn

)

+α2N−1

(1

hN−2+ 1

hN−1 + hN

)

+ α2N

(1

hN−1+ 1

hN

))

.

Hypothesis 7.1 yields

1

h0+ 1

h1≤ 1

2d1(1 + c0γ ),

1

h0 + h1+ 1

h2≤ 1

3d2

(c0γ (1 + γ ) + 2 + c0γ

2

1 + γ

),

1

hn−1+ 1

hn≤ 2

3dn(1 + c0γ ),

1

hN−2+ 1

hN−1 + hN≤ 1

3dN−1

(c0γ (1 + γ ) + 2 + c0γ

2

1 + γ

),

1

hN−1+ 1

hN≤ 1

2dN(1 + c0γ ).

Then (7.19) follows readily from the fact that for γ > 1,

c0γ (1 + γ ) + 2 + c0γ2

1 + γ> 2(1 + c0γ ).

Comparing with (7.8) and the proof of Lemma 7.2, we see that a direct estimate of

π j (�)′ relies on a bound for |D− 3

2 b|, which we can hardly expect to be sharp. Thisdifficulty can be bypassed by using the fact that � belongs to H1

0 (�)2 and arguingas in [23]. Uniqueness of the solution of (4.24) in X j implies that π j ( I (�)) = I (�)where I denotes the standard Lagrange interpolant in X j , that is well-defined because� is a line segment. This permits to write

π j (�)′ = π j (� − I (�))′ + ( I (�) − �)′ + �

′. (7.20)

First, we easily derive that

| I (�) − �|H1(�)≤ 2|�|H1(�)

.

Therefore

|π j (�)|H1(�)≤ 3|�|H1(�)

+ |π j (� − I (�))|H1(�), (7.21)

and it suffices to derive a bound for this last term. Let Mα = b be the system (4.24)for a generic component of � − I (�). Applying (7.12) and the analogue of (7.8), weobtain

123


|π j (� − I (�))|H1(�)≤ ‖(I + D− 1

2 K D12)−1‖2|D− 3

2 b| ≤ C |D− 32 b|, (7.22)

with the constant C of (7.12), provided c0 and γ are chosen as in the proof of Propo-

sition 7.1. Here we use the fact that (7.15) is also valid for ‖D− 12 K r D

12 ‖2. It remains

to estimate |D− 32 b|.

Proposition 7.2 Let � belong to H10 (�)2. If Hypothesis 7.1 holds with the constants

c0 and γ chosen as in the proof of Proposition 7.1, then each component � of � satisfies

|D− 32 b| <

√30c|�|H1(�)

, (7.23)

where c depends only on the interpolant I . Therefore,

∀� ∈ H10 (�)2, |π j (�)|H1(�)

<(3 + √

30cC)|�|H1(�)

, (7.24)

with the constants c and C of (7.23) and (7.12).

Proof We sketch the proof. To simplify, set w = � − I (�). Arguing as in Lemma 7.2,

we recover a bound for |D− 32 b| by replacing dn by d3

n and � by w in (7.10). The keypoint here is that the properties of the interpolant I imply

‖w‖2L2(x0,x1)

≤ c(h2

0|�|2H1(x0,x1)+ h2

1|�|2H1(x1,x2)

),

for a constant c independent of N , with analogous expressions for the norms in the

other subintervals. The factors involving hn in |D− 32 b| can be bounded by applying

(7.11) with c0γ = 32 , as in the proof of Proposition 7.1. This gives

|D− 32 b|2 < 10c

(

|�|2H1(x0,x1)+ |�|2H1(x0,x2)

+N∑

n=1

|�|2H1(xn−1,xn+1)+ |�|2H1(xN ,xN+1)

+|�|2H1(xN−1,xN+1)

)

,

and (7.23) follows.

Finally, space interpolation between (7.13) and (7.24) gives

∀� ∈ H12

00(�)2, |π j (�)|H

12

00 (�)< C |�|

H12

00 (�), (7.25)

with a constant C independent of N .

123


7.1.2 Second example

In the first example, � and X j have the same dimension, but of course this is notnecessary. For the unique solvability of (3.8), it is sufficient that the set of degreesof freedom of � be an adequate subset of those of X j , sufficiently rich to guaranteea good approximation property of the Lagrange interpolant I . Let us briefly giveanother simple choice of � with roughly half as many degrees of freedom as in thefirst example. Let N be an odd integer, keep the same ϕ0 and ϕN+1 as in (7.1), andchoose

ϕ1(x)|[x0,x1] = 1

h0(x − x0), ϕ1(x)|[x1,x2] = 1

h1(x2 − x),

ϕN (x)|[xN−1,xN ] = 1

hN−1, (x − xN−1) ϕN (x)|[xN ,xN+1] = 1

hN(xN+1 − x),

ϕ2n(x)|[x2n−2,x2n ] = 1

h2n−2 + h2n−1(x − x2n−2),

ϕ2n(x)|[x2n ,x2n+2] = 1

h2n + h2n+1(x2n+2 − x), 1 ≤ n ≤ N

2,

(7.26)

extended by zero elsewhere. We take

X j =⎧⎨

⎩v ∈ H1

0 (0, 1)2; v(x) = v1ϕ1(x) +N/2∑

n=1

v(x2n)ϕ2n(x) + vN ϕN (x)

⎫⎬

⎭,

(7.27)

� =⎧⎨

⎩μ ∈ H1(0, 1)2; μ(x) =

(N+1)/2∑

n=0

μ(x2n)ϕ2n(x)

⎫⎬

⎭. (7.28)

With this choice, (4.24) is uniquely solvable and it can be checked that all the resultsestablished for the first example carry over to this second example, with differentconstants.

7.2 The 3-D case

In addition to restricting the mesh in 3-D (see Remark 4.1), the above proofs arecomplex because they apply to a situation where the matrix of the system is irreducible.The proofs are much easier, when (3.8) represents local and independent conditions,but in general, this can only be achieved by suitably modifying the discrete velocityspaces. In this section we present an example for which (3.8) holds by explicitlyconstructing the solution of (4.3). The correction satisfies suitable continuity boundsthat are needed to establish the stability estimate (3.25). We follow the approach ofBenBelgacem [12] who presents a local construction in 3-D by adding to the spaceXh

s, j in �s, j a stabilizing bubble function in each face T ′ of the trace T hj,�i j

of the

123


triangulation T hj on �i j and choosing for �H constant vectors on each face T ′. More

precisely, for each j , 1 ≤ j ≤ Ms , and for each T in T hj , let P(T ) denote a polynomial

space such as the mini-element or Bernardi–Raugel element, used in approximatingthe velocity of the Stokes problem with order one. For each face T ′ of T h

j,�i j, let T be

the element of T hj with face T ′, and define

bT ′ |T = λ1λ2λ3,

where λk , k = 1, 2, 3, denote the barycentric coordinates of the three vertices of T ′;it is extended by zero outside T . Then set

Xhs, j = {v ∈ C0(�s, j )

3; ∀T ∈ T hj , T not adjacent to �i j , v|T ∈ P(T ),

∀T ∈ T hj , T adjacent to �i j , v|T ∈ P(T ) + bT ′R3},

(7.29)

and choose

�H = {μH ∈ L2(�i j )3; ∀T ′ ∈ T h

j,�i j,μH |T ′ ∈ P3

0}. (7.30)

As bT ′ vanishes at all vertices of T , it does not change the approximating propertiesof P(T ). Note that �H does not satisfy Hypothesis 3.2 (3). Nevertheless a discreteKorn inequality holds in V h

s , see Propositions 7.3 and 7.4 below. With this choice, wecan show that (3.8) holds. Indeed, for v ∈ X , it is easy to see that

vh =∑

T ′∈T hj,�i j

vT ′bT ′ , where vT ′ = 1∫

T ′ bT ′

∫

T ′v

satisfies (3.8). We now define

chj,�i j

(v) =∑

T ′∈T hj,�i j

cT ′bT ′ , where cT ′ = 1∫

T ′ bT ′

∫

T ′

(�h

s (v)|�s,i − Sh(v)|�s, j

),

(7.31)

where �hs (v)|�s,i is computed in the preceding subsection by correcting Sh(v)|�s,i

with chi,�i j

(v) defined in (4.17).

Lemma 7.4 The correction chj,�i j

(v) defined by (7.31) satisfies (4.3) and there existsa constant C independent of h and the diameter of �s,i , �s, j , and �i j such that forall v in H1

0 (�)n

|chj,�i j

(v)|H1(�s, j )≤ C |v|H1(�s,i ∪�s, j )

, ‖chj,�i j

(v)‖L2(�s, j )≤ Ch|v|H1(�s,i ∪�s, j )

.

(7.32)

123


Proof For any face T ′ of T hj,�i j

, we can write

cT ′ = 1∫

T ′ bT ′

∫

T ′

(�h

s (v)|�s,i − v − (Sh(v)|�s, j − v)).

But on �i j ,

�hs (v)|�s,i = Sh(v)|�s,i + ch

i,�i j(v),

owing to the support of each correction. Therefore

cT ′ = 1∫

T ′ bT ′

∫

T ′

(ch

i,�i j(v) + (Sh(v)|�s,i − v) − (Sh(v)|�s, j − v)

) = A1 + A2 + A3.

Let T be the element of T hj adjacent to T ′. By arguing as in the proof of Lemma 4.1,

we easily derive

|A3bT ′ |H1(T ) ≤ C |v|H1(�T ), (7.33)

where �T is the macro-element of T hj required to define Sh(v) in T . The estimation

of A2 is similar, but more technical because T ′ belongs to T hj whereas Sh(v)|�s,i is

constructed on T hi . Following the argument used in the proof of Lemma 4.3, we derive

|A2bT ′ |H1(T ) ≤ C∑

�

|v|H1(�T�), (7.34)

where {T�} denote the set of elements of T hi that intersect T .

The bound for A1 follows the same lines. With the notation of Lemma 4.1,

A1 = 1∫

T ′ bT ′

∫

T ′

∑

k

ckbk,

where the sum runs over the indices k for which Ok ∩T ′ �= ∅. According to Hypothesis4.1, the number of terms in this sum is bounded by a fixed integer. Thus

|A1| ≤∑

k

C ρ− 1

2k |v|H1(Dk )

,

and Hypothesis 4.1 implies that

|A1bT ′ |H1(T ) ≤ C |v|H1(�T�), (7.35)

123


where �T�is a macro-element in �s,i and again the number of elements in �T�

isbounded by a fixed integer. All constants are independent of h and the diameter of�s,i , �s, j , and �i j . Finally, the first inequality in (7.32) follows readily by summing(7.33)–(7.35) and applying Hypothesis 4.1. The argument for the second inequality in(7.32) is similar.

7.2.1 A discrete Korn inequality in V hs

Since all assumptions except Hypothesis 3.2 (3) hold, it suffices to examine the jumpof functions of V h

s through the interfaces �i j ∈ �ss . The next two lemmas show thatthe projection of this jump on polynomials of P1 is small.

Lemma 7.5 Let �i j ∈ �ss , i < j . There exists a constant C, independent of h andthe diameters of �i j , �s,i , and �s, j such that

∀vh ∈ V hs ,∀ p ∈ P3

1,

∣∣∣∣∣∣∣

∫

�i j

[vh] · p

∣∣∣∣∣∣∣≤ Ch

32 |�i j | 1

2(|vh |H1(Ds,i )

+ |vh |H1(Ds, j )

),

(7.36)

where Ds,k denotes the layer of elements in �s,k adjacent to �i j .

Proof All constants in this proof are independent of h and the diameters of �i j , �s,i ,and �s, j . Let T ′ be an element (i.e. a face) of T h

j,�i j, which is the mesh of �H . There is

no loss of generality in assuming that T ′ lies on the x1 − x2 plane. Consider a genericcomponent vh of vh . Since the jump already satisfies

∫

T ′[vh] = 0,

it suffices to bound the product of the jump by xk , k = 1, 2, say x1. Then for anyconstant c,

∫

T ′[vh]x1 =

∫

T ′

([vh] − c)⎛

⎝x1 − 1

|T ′|∫

T ′x1

⎞

⎠ .

A scaling argument gives

∥∥∥∥∥∥

x1 − 1

|T ′|∫

T ′x1

∥∥∥∥∥∥

L2(T ′)

≤ C |T ′| 12 hT ′ .

Similarly,

infc∈R ‖[vh] − c‖L2(T ′) ≤ ChT ′ |[vh]|H1(T ′) ≤ ChT ′

(|vh |�s,i |H1(T ′) + |vh |�s, j |H1(T ′)).

123


On one hand, equivalence of norms gives

|vh |�s, j |H1(T ′) ≤ C ρ− 1

2T |vh |�s, j |H1(T ),

where T is the element of T hj adjacent to T ′. On the other hand, arguing as in the

proof of Lemma 7.4, we prove that

|vh |�s,i |H1(T ′) ≤ C ρ− 1

2T |vh |�s,i |H1(�T ),

where �T is the union of all elements of T hi that intersect T ′. Then taking the product

and summing over all faces T ′ of T hj,�i j

, we obtain

∣∣∣∣∣∣∣

∫

�i j

[vh]x1

∣∣∣∣∣∣∣≤ Ch

32

⎛

⎜⎜⎝

∑

T ′∈T hj,�i j

|T ′|

⎞

⎟⎟⎠

12

(|vh |H1(Ds,i )+ |vh |H1(Ds, j )

),

whence (7.36).

Lemma 7.6 Let �i j ∈ �ss , i < j , and let �([vh]) ∈ P31 denote the projection of [vh]

onto P31 on �i j . There exists a constant C, independent of h and the diameters of �i j ,

�s,i , and �s, j such that

∀vh ∈ V hs , ‖�([vh])‖L2(�i j )

≤ Ch32(|vh |H1(Ds,i )

+ |vh |H1(Ds, j )

). (7.37)

Proof By definition �([vh]) ∈ P31 solves

∀ p ∈ P31,

∫

�i j

�([vh]) · p =∫

�i j

[vh] · p.

By passing to the reference subdomain � j , this reads

∀ p ∈ P31,

∫

�

�([vh]) ◦ Fj · p = |�| |�i j |−1∫

�i j

[vh] · p.

The coefficients of the polynomial of degree one �([vh]) ◦ Fj solve a linear systemwhose matrix only depends on �. Therefore, in view of (7.36),

‖�([vh])‖L2(�i j )≤ |�i j | 1

2 ‖�([vh]) ◦ Fj‖L∞(�)≤ Ch

32(|vh |H1(Ds,i )

+ |vh |H1(Ds, j )

).

This last result enables us to prove Korn’s inequality in V hs .

123


Proposition 7.3 Let |�s | > 0 and �s be connected. Then there exists h0 > 0 suchthat for all h ≤ h0,

∀vh ∈ V hs ,

Ms∑

i=1

|vh |2H1(�s,i )≤ C

Ms∑

i=1

‖D(vh)‖2L2(�s,i )

; (7.38)

the constants C and h0 are independent of h and the diameters of the interfaces �i j

and the subdomains �s,k .

Proof Let vh ∈ V hs . Formula (1.12) in [16] gives, with a constant C1 that depends

only on the shape regularity of the mesh of subdomains T�,

Ms∑

i=1

|vh |2H1(�s,i )

≤ C1

⎛

⎝Ms∑

i=1

‖D(vh)‖2L2(�s,i )

+∑

i< j

1

diam(�i j )‖�([vh])‖2

L2(�i j )

⎞

⎠

≤ C1

⎛

⎝Ms∑

i=1

‖D(vh)‖2L2(�s,i )

+ C∑

i< j

1

diam(�i j )h3(|vh |2H1(�s,i )

+ |vh |2H1(�s, j )

)⎞

⎠ ,

owing to (7.37). But h < diam(�i j ) and there exists an h0 > 0 such that

∀h ≤ h0, h2 (C1C)−1 ≤ 1

2.

Since C1 and C are independent of h and the diameters of the interfaces �i j and thesubdomains, then so is h0. Hence for all h ≤ h0,

Ms∑

i=1

|vh |2H1(�s,i )≤ 2C

Ms∑

i=1

‖D(vh)‖2L2(�s,i )

.

By arguing as in the proof of Lemma 3.4, we easily derive Korn’s inequality when|�s | = 0 and �s is connected.

Proposition 7.4 Let |�s | = 0 and �s be connected, i.e. �sd = ∂�s . Then there existsh0 > 0 such that for all h ≤ h0,

∀vh ∈ V hs ,

Ms∑

i=1

|vh |2H1(�s,i )≤ C

Ms∑

i=1

⎛

⎜⎝‖D(vh)‖2

L2(�s,i )+⎛

⎜⎝

2∑

l=1

∫

�sd

|vhs · τ l |

⎞

⎟⎠

2⎞

⎟⎠ ; (7.39)

123


the constants C and h0 are independent of h and the diameters of the interfaces �i j

and the subdomains �s,k .

The case when �s is not connected follows from these two propositions applied toeach connected component of �s according that it is or not adjacent to �s .

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