Technische Universit at Munchen, Boltzmannstraˇe 3, 85748 ...€¦ · [ANS01, ASW96, BS01] or...

ENERGY-CORRECTED FINITE ELEMENT METHODS

FOR THE STOKES SYSTEM

LORENZ JOHN∗, PETRA PUSTEJOVSKA∗, ULRICH RUDE†, BARBARA WOHLMUTH∗

∗Technische Universitat Munchen, Boltzmannstraße 3, 85748 Garching bei Munchen

†Universitat Erlangen-Nurnberg, Cauerstraße 11, 91058 Erlangen

Abstract. The solution of elliptic partial differential equations in a two-dimensional domain withre-entrant corners typically lacks full H2-regularity, and standard finite element approximations

on uniformly refined meshes do not, in general, show optimal order convergence in suitablyweighted L2-norms. Energy-corrected finite element techniques are based on a simple butparameter dependent local modification of the stiffness matrix. If applied properly, optimal order

convergence in weighted L2-norms can be recovered and the pollution effect is eliminated. Here,we generalize the ideas to the Stokes system and show that by using more than one correctionparameter optimal order convergence can be obtained.

1. Introduction

Flow problems on domains with re-entrant corners appear in many applications of interest.A typical example is the Stokes flow over a backward-facing step. It is well-known that in thepresence of re-entrant corners in the simulation domain with interior angle π < ω < 2π, the Stokessolution will, in general, have singular components even when the data are smooth. Therefore,standard finite element approximations show poor convergence results compared to the nodalinterpolation. More precisely, the so-called pollution effect, see, e.g., [Blu90], can be observed inboth, the standard L2- and also the weighted L2-norm. For early work on the regularity theory ondomains with non-convex corners, we refer to [Dau89, Gri85, KMR97, Kon67, OS95]. More recentresults can be found in [Cho14, GS06, KMR01, MS08] and the references therein.

Classical approaches to improve global convergence are graded meshes algorithms, see, e.g.,[ANS01, ASW96, BS01] or enrichment of the finite element space by singular components, see,e.g., [CK01, LP09]. Both these techniques improve the approximation quality of the discretespace but require a modification of the standard finite element scheme in an O(1) neighborhoodof the re-entrant corner. Here we follow a different idea. Quite often the quantity of interestcan be well approximated by the standard finite element space but the standard finite elementsolution is affected by the pollution effect. In these situations, energy-corrected schemes, see[Apo85, HHR+15, Rud89, RZ86, RZ92], do not enrich the finite element spaces associated with asequence of uniformly refined meshes and thus are quite attractive. The basic idea was originallyintroduced for finite difference schemes in [ZG78]. In recent contributions [ERW14, RWW14], amathematically rigorous analysis for the Laplace operator was presented. Although the techniquesare similar, the analysis for the Stokes system is more challenging since the structure of thesingular components is more complex, saddle point techniques must be taken into account, and themodification possibly depends on more than one parameter. Let us here also mention that thetheory below is derived for the norms of the weighted spaces, nevertheless, the optimal convergencecan be restored for other quantities of interest, like eigenvalues or stress intensity factors.

E-mail address: john,petra.pustejovska,[email protected], [email protected] words and phrases. asymptotic expansion, corner singularities, energy-corrected finite element methods,

pollution effect, re-entrant corners, Stokes system .

1

2 L.JOHN, P. PUSTEJOVSKA, U. RUDE AND B. WOHLMUTH

The rest of this paper is organized as follows: In Sec. 2, we state the model problem andthe abstract form of our energy-corrected finite element method for the Stokes problem. Sec. 3collects several preliminary results and Sec. 4 focusses on a priori convergence of the smoothremainders. Sec. 5 will be more technical, due to the symmetrization technique that is necessaryfor vector-valued finite element approximations. Here, we show that also a combination of singularfunctions does not contribute to the pollution in the approximation error. In Sec. 6, we proveour main theorem. Sec. 7 is devoted to the explicit construction of the modification. Definingthe correction parameters as roots of an energy defect function, we can show that the abstractconditions for optimality are satisfied. Finally, we present in Sec. 8 several numerical examplesfor different stable and stabilized formulations of the Stokes equations, as well as for differentcomputational geometries, illustrating the obtained theoretical results.

2. Energy-corrected finite elements for the Stokes system

In the following, we study the Stokes problem in the polygon Ω ⊂ R2 with a single re-entrantcorner and homogeneous Dirichlet boundary conditions:

(2.1)

−∆u +∇p = f in Ω,

div u = 0 in Ω,

u = 0 on ∂Ω,

where u is the velocity field, p the pressure and f represents a given force term. The corner isplaced at the origin, and r(x) = (

∑2i=1 x

2i )

1/2 is the distance of x to the non-convex corner. Wepoint out that our approach can be easily generalized to multiple re-entrant corners and is notrestricted to this type of boundary condition. However, for simplicity of notation, we will onlyconsider this situation.

Of crucial role for the regularity results of the solution are weighted Sobolev spaces. As usual,we denote by L2(Ω) and Hk(Ω), k ∈ R+, the Lebesgue and Sobolev–Slobodetskiı spaces withLebesgue index 2, and by ‖ · ‖0 and ‖ · ‖k their norms, respectively. For the characterization of thesolution regularity, we will use the weighted spaces, see, e.g., [Kuf85, AF03]. For an open, boundeddomain Ω ⊂ R2 and β ∈ R, we write L2

β(Ω), for the set of all measurable functions f : Ω → R,such that

‖f‖L2β(Ω) = ‖f‖0,β :=

(ˆΩ

|r(x)βf(x)|2 dx)1/2

<∞.

The function r2β is called the power weight, and L2β(Ω) stands for the associated weighted Lebesgue

space. For compatibility reasons, we also adopt the standard notation L20,β(Ω) for all functions

from L2β(Ω) with

´Ωf dx = 0. The weighted Sobolev space of Kondratev type are defined for

k ∈ N0 as follows:

Hkβ (Ω) :=

f ∈ L2

β(Ω) : rβ−k+|α|Dαf ∈ L2(Ω), |α| 6 k,

where α is a multi-index with |α| =∑2i=1 αi, and Dα represents the α-th generalized derivative,

see, e.g., [Kon67]. The space Hkβ (Ω) is equipped with the norm

‖f‖2Hkβ (Ω) = ‖f‖2k,β :=∑|α|6k

‖rβ−k+|α|Dαf‖20.

Note, that due to the special form of the weight r2β , L2β(Ω) and Hk

β (Ω) are Hilbert spaces. We also

write X2 for a space of vector valued functions which components belong to the function space X.Since we treat a model problem with Dirichlet boundary condition, we denote by Hk

0,β(Ω)2 the set

of all functions from Hkβ (Ω)2 having a zero trace. For a polygonal sub-domain Ωk ⊂ Ω, k ∈ N, we

define Hk−1/2β (∂Ωk)2 as a space of Dirichlet traces of functions from Hk

β (Ω)2, equipped with thenorm

‖g‖Hk−1/2β (∂Ωk)

:= infG|∂Ωk

=g‖G‖Hkβ (Ωk).

ENERGY-CORRECTED FINITE ELEMENT METHODS FOR THE STOKES SYSTEM 3

Through the paper, we use the symbols . and & for 6 c and > c, respectively, where c is a genericconstant, independent of the mesh-size parameter.

Although typical (H2, H1)-regularity of the solution (u, p) in the case of a domain with are-entrant corner cannot be expected, the regularity in weighted spaces, as showed in [BR81] forf ∈ L2(Ω)2 (see also for example [ANS01]), is higher. For f ∈ L2

β(Ω)2, we refer the reader to [GS06]where the result is stated also for a domain with multiple corners.

Lemma 1. Let Ω ⊂ R2 be a polygonal domain with a single re-entrant corner with angle ω ∈ (π, 2π),and let

β = 1 + ε− λ1,(2.2)

for any arbitrary small ε > 0, where λ1 depending on ω is the smallest positive solution of

sin(λω) = −λ sin(ω).

Then, for f ∈ L2β(Ω)2 and g ∈ H3/2

β (∂Ω)2, the unique solution of the Stokes problem (2.1) also

satisfies: (u, p) ∈ H2β(Ω)2 ×H1

β(Ω) and the a priori estimate

‖u‖2,β + ‖p‖1,β . ‖f‖0,β + ‖g‖H

3/2β (∂Ω)

.(2.3)

Before we introduce our modified Galerkin approach, let us specify our assumptions on the finiteelement spaces. The following theory will be constructed for a conforming, stable, low order finiteelement approximation Vh×Qh which is associated with a sequence of uniformly refined admissibleand shape-regular triangulations Th of Ω. A prototypical example is the P1–isoP2/P1 finite elementwhich is defined by

Vh =vh ∈ H1

0 (Ω)2 : vh|T ∈ P1(T ) ∀T ∈ Th/2,

Qh =qh ∈ L2

0(Ω) ∩ C(Ω) : qh|T ∈ P1(T ) ∀T ∈ Th,

where Th/2 is obtained by one uniform refinement of Th.It is well known, see, e.g., [Blu90] for the Stokes problem, that the accuracy of the standard

numerical approximation is significantly reduced due to the influence of the corner singularities, andthe reduction is propagated into the interior of the computational domain where the solution itselfis smooth. Namely, the standard numerical scheme exhibits, in general, (see [GR86, Sec.II.1.1.] or[ABF84]):

‖∇(u− uh)‖0 + h−λ1‖u− uh‖0 + ‖p− ph‖0 = O(hλ1).(2.4)

This behavior is called pollution effect, and from the structure of (2.4) it is clear that a simplechange of the error norm cannot reproduce the second order convergence of the method. Thisis also well documented by numerical experiments in Sec. 8. While the weighted L2

β(Ω)-normhelps to retrieve the convergence rates for the gradients of the velocity error, the velocity error inthe L2

β(Ω)-norm does not improve qualitatively although the best approximation error exhibits asecond order convergence rate.

Following the ideas of [ZG78, ERW14] for the Laplace operator, we now define the modifiedGalerkin approximation by: Find (um

h , pmh ) ∈ Vh ×Qh such that

(2.5)ah(um

h ,vh) + b(vh, pmh ) = 〈f ,vh〉 ∀vh ∈ Vh,

b(umh , qh) = 0 ∀qh ∈ Qh,

where the mesh-dependent bilinear form ah(·, ·) is defined for all v,w ∈ H1(Ω)2 by:

ah(w,v) := a(w,v)− dh(w,v) :=

ˆΩ

∇w : ∇v dx− dh(w,v),

and b(v, q) := −´

Ωq div v dx. The bilinear form ah(·, ·) differs from the one in the standard

Galerkin approximation in the local modification dh(·, ·) which is supported in the neighborhoodSh of the singular point with |Sh| = O(h2). More precisely, for a fixed κ > 0 we define:

Sh := int(∪T∈ThT ; dist(0, T ) 6 κh

).(2.6)


The value of κ will be specified later and as we will see, the set Sh will be often two layers ofelements around the singular point. As usual, we shall also require coercivity and continuity of theparticular bilinear forms, this means we assume the following additional conditions on dh(·, ·)

(A1) ellipticity of ah(·, ·): a(v,v)− dh(v,v) & ‖v‖21 for all v ∈ H1(Ω)2,(A2) continuity of dh(·, ·): |dh(w,v)| . ‖∇w‖L2(Sh)‖∇v‖L2(Sh) for all w,v ∈ H1(Ω)2,

(A3) symmetry of dh(·, ·): dh(w,v) = dh(v,w) for all w,v ∈ H1(Ω)2,(Sh) symmetry of Sh: The initial coarse mesh is on the patch, being the union of all elements

having the re-entrant corner as vertex, symmetric in the angular direction.

Under the assumptions (A1)–(A3), it is obvious that (2.5) has a unique solution and that thefollowing modified Galerkin orthogonality holds:

(2.7)a(u− um

h ,vh) + dh(umh ,vh) + b(vh, p− pm

h ) = 0 ∀vh ∈ Vh,

b(u− umh , qh) = 0 ∀qh ∈ Qh.

In what follows, we introduce the necessary and sufficient condition for optimal order convergencein terms of the energy defect function of the discretization, defined as:

gh(u, p) := a(u− umh ,u− um

h )− dh(umh ,u

mh ) + 2b(u− um

h , p− pmh ).(2.8)

As we will see later, this function reflects the pollution in the L2−approximation error, and thus,it is also called pollution function. The name energy defect function is motivated from the case ofthe incompressible homogeneous Dirichlet problem where the defect function can be rewritten as

gh(u, p) = a(u,u)− ah(umh ,u

mh ),(2.9)

Since the correction will be constructed in such a way that energy defect measured by gh(·, ·) iscontrolled, we speak about energy-corrected finite element schemes.

Now, we are ready to formulate a necessary condition for optimal order convergence.

Lemma 2 (Necessary condition). Let f ∈ L2−β(Ω)2 with β given by (2.2), and let ‖u− um

h ‖0,β =

O(h2). Then

gh(u, p) = O(h2).

Proof. Using the incompressibility of u and (2.9), we directly obtain:

gh(u, p) = a(u,u)− ah(umh ,u

mh ) 6 |〈f ,u〉 − 〈f ,um

h 〉| 6 ‖f‖0,−β‖u− umh ‖0,β .

The conclusion follows by the assumption of the lemma.

Before we can formulate a sufficient condition in terms of gh(·, ·), we have to consider the structureof the solution (u, p) in more detail. Also, we will often use formulations in polar coordinates (r, θ).For convenience of the reader, we recall some results from [Kon67, Gri85, Dau89, OS95] which areessential for our study. Since the eigenfunctions of the Stokes operator at the singular corner areknown, the solution (u, p) can be decomposed into singular components and a smooth remainder.The singular components are characterized by the power exponents λi which have to satisfy in thecase of homogeneous Dirichlet boundary conditions the following eigenvalue equation:

λ2i sin2(ω)− sin2(λiω) = 0, λi 6= 0, λi ∈ C,(2.10)

dependent on the non-convex angle ω, see ,e.g., [OS95, KMR97]. One can show, that the roots of(2.10) have a maximal algebraic (and thus also geometric) multiplicity of two, see [KMR01]. Foreach eigenvalue λi with geometric multiplicity 1 6 Ii 6 2, we denote the θ-part of the i-th singularsolution by (

ϕ(i)j,k(θ), ψ

(i)j,k(θ), ξ

(i)j,k(θ)

)>j=1,...,Iik=0,...,κij−1

.(2.11)

Here κij 6 2 stands for the length of the Jordan block of the associated jth eigenfunction,

1 ≤ j ≤ Ii, and ϕ(i)j,k(θ), ψ

(i)j,k(θ), ξ

(i)j,k(θ), k = 0, . . . κij − 1, form a chain of eigensolutions satisfying

the homogeneous Dirichlet boundary conditions. Without specification of (2.11), we can recall the


following decomposition theorem for the Stokes problem (2.1), see [OS95, Theorem 4.1] and theoriginal work [Kon67].

Lemma 3. Let α, β be positive real numbers and Ω be a polygon with a single re-entrant corner ofangle ω ∈ (π, 2π). Let (u, p) ∈ H2

0,β(Ω)2 ×H1β(Ω)∩L2

0(Ω) be the weak solution of problem (2.1) for

a given f ∈ L2−α(Ω)2. Moreover, let us assume that the band

1− β < Re(λ) < 1 + α(2.12)

contains N roots λ1, ..., λN of equation (2.10). Additionally, let us assume, that there are no rootsof (2.10) with Re(λ) = 1 + α and Re(λ) = 1− β. Then, near the singular point, there holds thefollowing expansion of the solution:

u =

N∑i=1

Ii∑j=1

κi,j−1∑k=0

ci,j,k sui,j,k + U, p =

N∑i=1

Ii∑j=1

κi,j−1∑k=0

ci,j,k spi,j,k + P,(2.13)

where ci,j,k are constants. The singular parts of the decompositions can be represented in polarcoordinates as

sui,j,k = η(r) rλik∑l=0

1

l!(ln r)l

(ϕ

(i)j,k−l(θ)

ψ(i)j,k−l(θ)

), spi,j,k = η(r) rλi−1

k∑l=0

1

l!(ln r)lξ

(i)j,k−l(θ),

where the smooth cut-off function η(r) is equal to one in an O(1) neighborhood of the origin, andthe regular parts satisfy:

(U, P ) ∈ H2−α(Ω)2 ×H1

−α(Ω).

The coefficients ci,j,k and the regular parts (U, P ) also satisfy the stability condition:

N∑i=1

Ii∑j=1

κi,j−1∑k=0

|ci,j,k|+ ‖U‖2,−α + ‖P‖1,−α . ‖f‖0,−α.

We call sui,j,k, spi,j,k the singular functions and U, P the smooth remainders. For simplicity ofnotation, we neglect the index j and k if j = 1 and k = 0. Note also, that for the decomposition(2.13), a higher regularity of the right-hand side f is required, as in comparison with the regularityresult from Lemma 1. We also include a graph, Fig. 1, of the real part of the solutions λi of (2.10)in relationship to ω. The roots λi of an increased multiplicity are exactly those roots which lay onthe bifurcations of the depicted curves, and, the case when λ2 = 1 (see discussion below).

For ω = 3/2π, a problem widely used in engineering applications, we specify the first twosingular functions, i = 1, 2 (note that the associated algebraic multiplicity of λi are equal to one),in the polar coordinates:

(sui )r(sui )θspi

= η(r) rλi

ϕ(i)

ψ(i)

ξ(i)

= η(r) rλi

ci1sin[(λi + 1)θ]

cos[(λi + 1)θ]0

+ ci2

− cos[(λi + 1)θ]sin[(λi + 1)θ]

0

+ci3

(λi − 1) cos[(λi − 1)θ]−(λi + 1) sin[(λi − 1)θ]

4λir cos[(λi − 1)θ]

+ ci4

(λi − 1) sin[(λi − 1)θ](λi + 1) cos[(λi − 1)θ]

4λir sin[(λi − 1)θ]

,where ci1 = (λ2

i − 1) sin(λiω), ci2 = −(λi− 1)λi cos(λiω), ci3 = −λi cos(λiω), ci4 = −(λi− 1) sin(λiω)are specified by the Dirichlet boundary conditions and λ1 = 0.54448.., λ2 = 0.90852...

The decomposition (2.13) allows us to formulate a sufficient condition for second order conver-gence in the velocity and first order convergence in the pressure in a suitable weighted L2-norm.

Theorem 4 (Sufficient condition). Let (A1)–(A3) hold, ω ∈ (π, ω3), ω 6= ω2, and f ∈ L2−α(Ω)2 for

some α such that 1− λ1 < α < min1,Re(λ3)− 1. Additionally, let (Sh) hold if ω ∈ [ω1, ω3), and

gh(sui , spi ) = O(h2), for

i = 1 if ω ∈ (π, ω2),i = 1, 2 if ω ∈ (ω2, ω3),

(2.14)


0 π ω1 ω3ω2 32π 2π

0.5

1

2

3

Re(λ)

Figure 1. Distribution of the real part of the eigenvalues λ with respect to theangle ω of the re-entrant corner. The blue lines represent complex eigenvalues,the orange ones represent the real eigenvalues. The points of bifurcations, alsothe point where λ2 = 1, have increased multiplicity. In the horizontal axis, ωi,i = 1, 2, 3, represent the following cases: ω1 is defined such that λ1 + λ2 = 2,i.e., ω1 = (1.22552...) · π, ω2 is the unique angle such that λ2 = 1, i.e., ω2 =(1.430296...) · π, and ω3 represents the angle for which λ1 + Re(λ3) = 2, i.e.,ω3 = (1.64941...) · π.

where ω2, ω2 and ω3 are defined in Fig 1. Then, the modified Galerkin approximation (umh , p

mh ) ∈

Vh ×Qh converges optimally, i.e.,

‖u− umh ‖0,α . h2‖f‖0,−α, and ‖u− um

h ‖1,α + ‖p− pmh ‖0,α . h‖f‖0,−α.(2.15)

Note, that the sufficient condition for (2.15) is formulated in terms of the pollution function forthe singular functions. This means, that these are the components of the whole gh(u, p), whichare responsible for the reduced convergence rate of the standard scheme. This will be more clearlater, as we explicitly show the expansion of gh(u, p) with the help of the decomposition (2.13).Since Theorem 4 is also formulated for different cases of the non-convex angle ω, let us now brieflyexplain its limits ω1, ω2 and ω3, see also Fig. 1.

i) ω ∈ (π, ω1). The condition on local symmetry of the mesh can be in this case dropped, becausethe second convergence order of the combination of the first and second singular functionsin the expansion of gh(u, p) follows directly by standard approximation and interpolationestimates. On the other hand, for ω > ω1 we shall use some symmetrization techniques, whichwill require a locally symmetric mesh around the singular point.

ii) ω ∈ (π, ω2). In this case, the second singular functions (su2 , sp2) are already in H2(Ω)2×H1(Ω))

(easy to see due to the regularity of su2 ∼ rλ2 and λ2 > 1), and thus, only the pollution createdby the first singular functions contributes to the pollution of the whole solution.

iii) ω ∈ (ω2, ω3). Optimal convergence of the approximations is recovered if the pollution of thefirst two singular functions is eliminated.

iv) ω ∈ [ω3, 2π). In this case, Re(λ3)− 1 6 1− λ1 < α and thus the decomposition of the solutionhas to have three or four singular functions, see (2.12). We shall not consider this case inour analytic study since the key-stone of the proof of Theorem 4 is the symmetry argumentbetween su1 and su2 and their discrete counterparts. Such property is on the other hand notusable for the (su1 , s

u3 ) pairing, rather general orthogonality and its conversion on the discrete

level has to be studied. This, however, exceeds the framework of this work. Let us mentionhere, that the numerical experiments suggest that condition (2.14) could be sufficient for theoptimality result (2.15) also for angles larger than ω3, if the local mesh around the singularpoint satisfies some additional uniformity.


v) ω = ω2. As one can see from Fig. 1, λ1 and λ2 are simple roots of (2.10) for all ω ∈ (π, ω3)and ω 6= ω2. This means that the decomposition (2.13) has a simplified form

u =

2∑i=1

ci sui + U, p =

2∑i=1

ci spi + P.

On the other hand, the case of ω = ω2 has a special property, coming from the fact that λ2 = 1is a root of (2.10) with double algebraic and single geometric multiplicity. The decomposition(2.13) (for simplicity presented only for the velocity) thus takes the form:

u = c1su1 + c2,1,0 su2,1,0 + c2,1,1 su2,1,1 + U,

where the singular components are

su2,1,0 = η(r) r

(ϕ

(2)1,0(θ)

ψ(2)1,0(θ)

)and su2,1,1 = η(r) r

[(ϕ

(2)1,1(θ)

ψ(2)1,1(θ)

)+ ln(r)

(ϕ

(2)1,0(θ)

ψ(2)1,0(θ)

)].

In our study, we shall exclude this case mainly due to the reasons of readability of the technicalresults.

In order to obtain the estimates (2.15), we shall rely on the linearity of the system, which meansthat we can split the error due to (2.13) into:

u− umh =

2∑i=1

cui (sui − suimh ) + U−Um

h , p− pmh =

2∑i=1

cui (spi − spi

mh ) + P − Pm

h ,

where (suimh , s

pi

mh ) and (Um

h , Pmh ) represent the modified finite element approximations of the

singular functions (sui , spi ) and smoother remainders (U, P ), respectively (also solutions of the

Stokes problem with special data). The proof of Theorem 4 is postponed to Section 6. It dependson a series of preliminary results which are provided in the next sections.

3. Preliminary results

In the analysis below, we will often use the interpolation results of the standard nodal interpolants,namely:

Lemma 5 (Interpolation errors in weighted Sobolev spaces). Let Iuh : C(Ω)→ Vh, Iph : C(Ω)→ Qhbe standard nodal interpolation operators, and let v ∈ H2

α(Ω)2 and q ∈ H1α(Ω) for some α < 1.

Then, for β such that α− 1 6 β 6 α, and for k = 0, 1, we have

‖∇k(v − Iuhv)‖0,β . h2−k−α+β ‖v‖2,α, ‖q − Iphq‖0,β . h1−α+β‖q‖1,α.(3.1)

The proof can be found in [BD82] or [Orl98], and it is based on standard scaling arguments.For the weighted spaces, we can also derive an equivalent to the inverse inequality. Its proof

follows also directly from standard scaling arguments, hence we do not include it here.

Lemma 6 (Inverse estimates in weighted Sobolev spaces). For any vh ∈ Vh and β > −1 thereholds

‖vh‖1,β . h−1 ‖vh‖0,β .(3.2)

As mentioned before, the pollution effect is reflected only in the L2(Ω) error estimates. Thismeans, that the modified and standard Galerkin approximations share the same approximationproperties in the energy space for the velocity, H1(Ω)2. More precisely we have the following:

Theorem 7. Let (A1)–(A3) hold, and let u ∈ H2α(Ω)2 ∩H1

0 (Ω)2 and p ∈ H1α(Ω)∩L2

0(Ω) for someα such that 1− λ1 < α < min1,Re(λ3)− 1. Then the following energy error estimates for themodified Galerkin approximation hold:

‖∇(u− umh )‖0 + ‖p− pmh ‖0 . h1−α(‖u‖2,α + ‖p‖1,α),

‖∇(sui − suimh )‖0 + ‖spi − s

pimh ‖0 . h

λi , i = 1, 2.


Proof. Using the standard Strang argument, cf. [SF73] or [BBF13], we obtain together with (2.7):

‖u−umh ‖1 + ‖p− pm

h ‖0

.

(inf

vh∈Vh

‖u− vh‖1 + infqh∈Qh

‖p− qh‖0 + sup0 6=vh∈Vh

ah(u,vh) + b(vh, p)− 〈f ,vh〉‖vh‖1

)

.

(inf

vh∈Vh

‖u− vh‖1 + infqh∈Qh

‖p− qh‖0 + sup0 6=vh∈Vh

dh(u,vh)

‖vh‖1

).(‖u− Iuhu‖1 + ‖p− Iphp‖0 + ‖∇u‖L2(Sh)

).

By the embedding H2α(Ω) → H1

α−1(Ω), see, e.g., [Kuf85], and the fact that r . h on Sh, we have

‖∇u‖L2(Sh) . h1−α‖u‖H1

α−1(Sh) . h1−α‖u‖2,α.(3.3)

This, together with (3.1) for β = 0, implies the first assertion. The second assertion for the singularfunctions can be derived similarly. Nevertheless, since sui 6∈ H2

1−λi(Ω)2 and spi 6∈ H11−λi(Ω), the

interpolation error from (3.1) would give us a slightly suboptimal estimate. However, a directcomputation of the interpolation error and of the gradient of sui on Sh lead to the desired result.

As last in this section, we turn our attention to the incompressibility constraint of the systemand its influence on the existence of the solution. It is known that for the well-posedness ofincompressible flow problems, the inf-sup condition plays a crucial role. Thus, let us formulate itsgeneralization to the weighted Sobolev spaces.

Lemma 8 (Continuous inf-sup in weighted spaces). Let q ∈ L20,β(Ω), for β such that −1 < β < 1.

There exists a constant c > 0 such that

sup0 6=v∈H1

0,−β(Ω)2

(div v, q)

‖∇v‖0,−β> c‖q‖0,β .(3.4)

Proof. The proof is based on the solvability of the divergence equation in weighted spaces. By theresults of [Hub11] (see also [DN90, DM01] for the power weights), there exists, for −1 < β < 1, abounded, linear operator B : L2

0,β(Ω)→ H10,β(Ω)2, called Bogovskiı operator, such that div(Bf) = f

for all f ∈ L20,β(Ω). Moreover, due to the continuity, there holds:

‖∇(Bf)‖0,β . ‖f‖0,β .(3.5)

Let us then for q ∈ L20,β(Ω) define w ∈ H1

0,−β(Ω)2 by div w = r2βq. Then, by (3.5) we have

‖∇w‖0,−β . ‖r2βq‖0,−β . ‖q‖0,β , and thus

‖q‖0,β .´

Ωr2βq2dx

‖q‖0,β.

(div w, q)

‖∇w‖0,−β. sup

0 6=v∈H10,−β(Ω)2

(div v, q)

‖∇v‖0,−β,

which proves the assertion.

In the proof of Theorem 4, we shall consider only stable finite element spaces (Vh, Qh) for which

exists a linear, H1−stable projection operator Πdivh : W 1,1

0 (Ω)2 → Vh preserving the divergencein the dual of the pressure space. In the case of P1–isoP2/P1 , we refer the reader to [Ver84],and to [BBDR12, BBF13] for explicit construction of Πdiv

h for the low order Mini-, P2–P0 or theTaylor–Hood element. Note also that the above mentioned properties of Πdiv

h are fulfilled by theFortin operator. By this assumption on the (Vh, Qh) pair, we obtain the discrete counterpart of(3.4).

Lemma 9 (Discrete inf-sup in weighted spaces). Let Vh×Qh be a stable finite element space pair,and let −1 < β < 1. Then, there exists a constant c > 0, independent of h, such that

sup0 6=vh∈Vh

(div vh, qh)

‖∇vh‖0,−β> c‖qh‖0,β for all qh ∈ Qh.(3.6)


Proof. One can show by suitable scaling arguments in weighted spaces, see [Cia02], that theprojection operator Πdiv

h is also globally H1β−stable, i.e.

‖Πdivh v‖1,β . ‖v‖1,β ,

and for v ∈ H2α(Ω) and α < 1, such that α − 1 6 β 6 α there holds a projection error estimate

analogous to (3.1):

‖∇k(v −Πdivh v)‖0,β . h2−k−α+β ‖v‖2,α, k = 0, 1.(3.7)

From Lemma (8) and the projection properties of Πdivh , we directly have that

‖qh‖0,β . sup06=v∈H1

0,−β(Ω)2

(div v, qh)

‖∇v‖0,−β. sup

0 6=v∈H10,−β(Ω)2

(div Πdivh v, qh)

‖∇Πdivh v‖0,−β

. sup0 6=vh∈Vh

(div vh, qh)

‖∇vh‖0,−β.

4. A priori convergence results in negatively weighted norms

In this section, we study the approximation error for the smooth remainders in the negativelyweighted norms. This will be essential for the duality arguments in the proof of the main theoremin Sec. 6, hence, both results, which we present here, are auxiliary ones. First, we show the optimalbounds for the standard Galerkin scheme, and in view of this result, we focus on the approximationerror for the modified finite element approximation. We would like to mention that both techniquesare rather standard, mainly based on the same duality argument and the generalization of theresults from [BR88]. But for the sake of completeness, due to the fact that the optimal bound forthe smooth functions in weighted norms is not, in general, widely used, we present them in twoseparate theorems in their full form.

Theorem 10. Let (A1)–(A3) hold, and let U ∈ H2−α(Ω)2 ∩H1

0 (Ω)2 and P ∈ H1−α(Ω) ∩L2

0(Ω) forsome 1− λ1 < α < min1,Re(λ3)− 1. Then,

‖∇(U−Uh)‖0,−α + ‖P − Ph‖0,−α . h (‖U‖2,−α + ‖P‖1,−α) ,(4.1)

‖U−Uh‖0,−α . h2(‖U‖2,−α + ‖P‖1,−α).(4.2)

Proof. We use the ideas from [BR88] for the Poisson equation and estimate the approximationerrors in the norms of weighted spaces with r2α weight by approximation errors in norms witha suitable mesh-dependent weight. For that, let us define the weight %2 := (r2 + θh2), for somesufficiently large but fixed positive θ. Then, generalizing the idea for the Stokes problem, usingthe technique of [DN90, GL12], one can derive corresponding approximation errors for standardGalerkin approximations (vh, qh) of v ∈ H2

0,−α(Ω)2 and q ∈ H1−α(Ω) ∩ L2

0(Ω), 0 < α < 1:

(4.3)‖%−α∇(v − vh)‖0 + ‖%−α(q − qh)‖0

. ‖%−α∇(v −Πdivh v)‖0 + h−1‖%−α(v −Πdiv

h v)‖0 + ‖%−α(q − Iphq)‖0.

More details on the mesh-dependent power weights and the error estimates in the respective spacescan be also found in [Cia02]. Since ‖ · ‖0,−α and ‖ρ−α · ‖0 are equivalent norms in the finite elementspace, see [BR88], we can directly by the triangle inequality and the fact that ρ−α < r−α write:

‖∇(U−Uh)‖0,−α + ‖P − Ph‖0,−α. ‖∇(U− IuhU)‖0,−α + ‖%−α∇(U−Uh)‖0 + ‖%−α∇(U− IuhU)‖0

+ ‖P − IphP‖0,−α + ‖%−α(P − Ph)‖0 + ‖%−α(P − IphP )‖0. ‖%−α∇(U−Uh)‖0 + ‖%−α(P − Ph)‖0 + ‖∇(U− IuhU)‖0,−α + ‖P − IphP‖0,−α.

The contributions with mesh-dependent weight %−α can be bounded with the help of (4.3), whichyields, together with %−α < r−α and the interpolation/projection error estimates (3.1) and (3.7),the first assertion (4.1).


The L2−α(Ω) estimate for the velocity is then derived by a duality argument. Let us consider

the following homogeneous Dirichlet problem:

−∆v +∇z = r−2α(U−Uh) in Ω,

div v = 0 in Ω,

v = 0 on ∂Ω.

Then by Lemma 1, we have that v ∈ H20,α(Ω)2 and z ∈ H1

α(Ω) ∩ L20(Ω), since r−2α(U −Uh) ∈

L2α(Ω)2, for α satisfying the assumption of the theorem. Using the standard Galerkin orthogonality

for approximation (Uh, Ph), the a priori estimate ‖v‖2,α + ‖z‖1,α . ‖r−2α(U − Uh)‖0,α =‖U−Uh‖0,−α, the interpolation error estimate (3.1) and (4.1), we obtain:

‖U−Uh‖20,−α = (U−Uh,−∆v) + (U−Uh,∇z) = a(U−Uh,v) + b(U−Uh, z)

= a(U−Uh,v − Iuhv) + b(v − Iuhv, P − Ph) + b(U−Uh, z − Iphz)6 ‖∇(U−Uh)‖0,−α‖∇(v − Iuhv)‖0,α + ‖∇(v − Iuhv)‖0,α‖P − Ph‖0,−α

+ ‖∇(U−Uh)‖0,−α‖z − Iphz‖0,α. h2(‖U‖2,−α + ‖P‖1,−α)(‖v‖2,α + ‖z‖1,α)

. h2(‖U‖2,−α + ‖P‖1,−α)‖U−Uh‖0,−α.

With this, we have completed the proof.

Now, we can establish the desired bound of the error of the smooth remainder.

Theorem 11. Let (A1)–(A3) hold, and let U ∈ H20,−α(Ω)2 and P ∈ H1

−α(Ω) ∩ L20(Ω) for some α

such that 1− λ1 < α < min1,Re(λ3)− 1. Then, the following error estimates are valid

‖U−Umh ‖0,−α . h2 (‖U‖2,−α + ‖P‖1,−α) ,

‖U−Umh ‖1,−α + ‖P − Pm

h ‖0,−α . h (‖U‖2,−α + ‖P‖1,−α) .

Proof. Since we already have the results for the standard Galerkin approximation (4.1) and (4.2),we directly start with the dual problem:

−∆v +∇z = r−2α(U−Umh ) in Ω,

div v = 0 in Ω,

v = 0 on ∂Ω.

Then, noting that U−Umh = 0 on ∂Ω and r−2α(U−Um

h ) ∈ L2α(Ω)2, the error can be rewritten

by the modified Galerkin orthogonality as:

(4.4)

‖U−Umh ‖20,−α = a(U−Um

h ,v) + b(U−Umh , z)

= a(U−Uh,v) + b(U−Uh, z) + ah(Uh −Umh ,v

mh ) + b(Uh −Um

h , zmh )

= a(U−Uh,v) + b(U−Uh, z)− dh(Uh,vmh ).

Using the a priori estimate for v, z:

‖v‖2,α + ‖z‖1,α . ‖r−2α(U−Umh )‖0,α . ‖U−Um

h ‖0,−α,(4.5)

we can bound the Ω-supported terms of (4.4) by Theorem 10:

a(U−Uh,v) + b(U−Uh, z) = (U−Uh,−∆v +∇z) 6 ‖U−Uh‖0,−α‖ −∆v +∇z‖0,α6 ‖U−Uh‖0,−α‖U−Um

h ‖0,−α. h2(‖U‖2,−α + ‖P‖1,−α)‖U−Um

h ‖0,−α.

The Sh-supported term dh(Uh,vmh ) can be bounded using Theorem 7 and Theorem 10 and the same

arguments as for (3.3). Then, by the definition of the weighted norm, we obtain by Theorem 10the following:

‖∇Uh‖L2(Sh) 6 hα‖∇(U−Uh)‖L2

−α(Sh) + ‖∇U‖L2(Sh) . h1+α(‖U‖2,−α + ‖P‖1,−α),(4.6)


component ∂∂r

∂∂θ

(su1 )r

(su1 )θ

sp1

Table 1. Schematic plots representing the symmetry properties of both com-ponents of su1 and sp1 for the case of ω = 3/2π, including plots of the particularderivatives in radial and angular directions.

and, by Theorem 7 −β = α,

‖∇vmh ‖L2(Sh) 6 ‖∇(v − vm

h )‖0 + ‖∇v‖L2(Sh) . h1−α (‖v‖2,α + ‖z‖1,α) .(4.7)

Combining (4.6) and (4.7), we obtain with the help of (4.5), the bound:

|dh(vmh ,Uh)| 6 ‖∇Uh‖L2(Sh)‖∇vm

h ‖L2(Sh) . h2(‖U‖2,−α + ‖P‖1,−α)‖U−Um

h ‖0,−α,

and thus the L2−α(Ω)−estimate.

The H1−α(Ω)2 estimate follows directly by splitting the error into the approximation and

interpolation error together with (3.1), (3.2) and the above proven L2−α(Ω)−estimate, namely:

(4.8)

‖U−Umh ‖1,−α . ‖U− IuhU‖1,−α + ‖IuhU−Um

h ‖1,−α. h‖U‖2,−α + h−1 (‖U− IuhU‖0,−α + ‖U−Um

h ‖0,−α)

. h(‖U‖2,−α + ‖P‖1,−α).

In the last step, we prove the estimate for the pressure using the discrete inf-sup condition (3.6)and the estimates (4.6) and (4.8). More precisely, for the interpolation IphP ∈ Qh, we have by (2.7)and the triangle inequality on ‖Um

h ‖H1−α(Sh):

‖IphP − Pmh ‖0,−α . sup

0 6=vh∈Vh

b(vh, IphP − Pm

h )

‖vh‖1,α. sup

0 6=vh∈Vh

b(vh, P − Pmh ) + b(vh, I

phP − P )

‖vh‖1,α

. sup0 6=vh∈Vh

−a(U−Umh ,vh)− dh(Um

h ,vh) + b(vh, IphP − P )

‖vh‖1,α. ‖U−Um

h ‖1,−α + ‖IphP − P‖0,−α + ‖U‖H1−α(Sh).

Using similar arguments as for (3.3), but this time for the embedding H2−σ+1(Ω)2 → H1

−σ(Ω)2,σ ∈ R, we obtain that

‖IphP − Pmh ‖0,−α . h (‖U‖2,−α + ‖P‖1,−α) + ‖IphP − P‖0,−α,

and thus, by the triangle inequality and the interpolation error (3.1) we finally obtain:

‖P − Pmh ‖0,−α . h (‖U‖2,−α + ‖P‖1,−α) + ‖P − IphP‖0,−α . h (‖U‖2,−α + ‖P‖1,−α) ,

by which we have finished the proof.


5. Symmetry arguments

Now, let us turn our attention to the modified approximation of the singular functions. We havealready mentioned that the necessary condition in Theorem 4 is formulated as a condition on thepollution of (sui , s

pi ). Nevertheless, as we will see in the next section, the pollution function gh(u, p)

defined in (2.8) contains also terms with combination of the singular functions. In this section, weshow that these parts of gh(u, p) converge though fast enough, even despite the reduced regularityof (sui , s

pi ).

The proof here is based on the idea of [ERW14] for the Poisson equation, where the symmetryproperties of the first and second singular functions are used. Nevertheless, due to the vectorvalued Stokes problem, its generalization is technical and requires additional estimates, we state atfirst an auxiliary lemma, where we construct another local finite element functions which help usto symmetrize the possibly non-symmetric suj

mh

and spjm

h.

Let us define the following finite element spaces:

VΩ1

h := vh ∈ H1(Ω1) : vh = vh|Ω1, vh ∈ Vh, QΩ1

h := qh ∈ L2(Ω1) : qh = qh|Ω1, qh ∈ Qh,

for a fixed domain Ω1 ( Ω with |Ω1| = O(1), matching the initial local symmetric grid (see Fig. 3,

left). We define an Ω1-mirror-reflected finite element function gu∗jh ∈ VΩ1

h and corresponding

pressure gp∗jh ∈ QΩ1

h by a local discrete Stokes problem:

(5.1)ah(gu∗jh ,vh) + b(vh, g

p∗jh) = ah(suj

m

h,vh) + b(vh, s

pj

m

h) for all vh ∈ VΩ1

h ∩H10 (Ω1),

bΩ1(gu∗jh , qh) = bΩ1

(sujm

h, qh) for all qh ∈ QΩ1

h ,

where bΩ1(vh, qh) :=

´Ω1qh div(vh) dx, and close the system by the condition on the pressure:

gp∗jh − spj

m

h∈ L2

0(Ω1), and the following Dirichlet boundary condition on the velocity gu∗jh :

(5.2)

(5.3)

gu∗jh(x) =

(− (suj

m

h)r(x

∗), (sujm

h)θ(x

∗))>

if (suj )r is anti-symmetric,((suj

m

h)r(x

∗),−(sujm

h)θ(x

∗))>

if (suj )r is symmetric,

where x∗ denotes a mirror symmetric point of x ∈ ∂Ω1 with respect to the axis (r, ω/2). A sketchdepicting the definition of the boundary condition for anti-symmetric (suj )r together with themirror-reflection of x is presented in Fig. 2.

Lemma 12. The discrete solution (gu∗jh , gp∗jh) of (5.1) with boundary condition (5.2), respectively

(5.3), exists and is unique as it satisfies the discrete compatibility conditionˆ∂Ω1

gu∗jh · n dx =

ˆ∂Ω1

sujm

h· n dx.

Additionally, it holds:

‖gu∗jh − sujm

h‖H1/2(∂Ω1\(∂Ω∩∂Ω1)) . ‖suj − suj

m

h‖H1/2(∂Ω1\(∂Ω∩∂Ω1)),(5.4)

and the sum sujmh

+ gu∗jh has the same symmetry properties in Ω1 as suj , and spjm

h+ gp∗jh as spj .

Proof. Let us in this proof consider the boundary condition (5.2) only, the latter case followssimilarly. Since (suj )r is in this case anti-symmetric, (suj )θ is symmetric and spj is anti-symmetric inthe angular directions, see the first column of plots in Table 1 for the case of j = 1 and a L-shapedomain.

Using the fact that the outer normal vector is reflected in the following way: n = (nr, nθ)> =

(n∗r ,−n∗θ)> (see Fig. 2, left), from the definition (5.2) it then follows thatˆ∂Ω1

gu∗jh(x) · n dx =

ˆ∂Ω1

−(sujm

h)r(x

∗)n∗r + (sujm

h)θ(x

∗)n∗θ rdr dθ =

ˆ∂Ω1

sujm

h· n dx.

Thus, the solution of (5.1) and (5.2) exists and is unique due to the mean value condition on gp∗jh.


nr

n∗r

nθ

n∗θ

Ω1

x

x∗ω2

ω0

(suj )r

(sujmh)r

(gu∗jh )r

(sujmh)r + (gu∗jh )r

Figure 2. (Anti-)symmetrization. Left: The mirror-reflected point x∗ to x ∈ ∂Ω1

across the axis (r, ω2 ), and the schematic representation of the outer normal

n = (nr, nθ)>, in the case of an L-shape domain. The black parts of ∂Ω1 represent

∂Ω ∩ ∂Ω1. Right: Exemplary profiles of the radial components of gu∗jh as defined

in (5.2) on ∂Ω1 \ (∂Ω∩ ∂Ω1) and of the sum sujmh

+ gu∗jh for a given anti-symmetric

radial component of suj , and sujmh

which is generally free of symmetry. In bothgraphics, the empty circles represent the vertices of the elements in Ω1 of a currentrefinement level.

The property (5.4) follows from definition (5.2), since we have:

(suj − gu∗jh)(x) =(− (suj )r(x

∗) + (sujm

h)r(x

∗), (suj )θ(x∗)− (suj

m

h)θ(x

∗))>,

and the application of the triangle inequality.Let us now focus on the symmetry property of the sum suj

mh

+ gu∗jh . For each x ∈ ∂Ω1 we havethen, as schematically depicted for the first component in Fig. 2 right:

(sujm

h+ gu∗jh)(x) =

((suj

m

h)r(x)− (suj

m

h)r(x

∗), (sujm

h)θ(x) + (suj

m

h)θ(x

∗))>,(5.5)

with explicit gu∗jh = 0 on ∂Ω∩∂Ω1 due to sujmh

= 0 on ∂Ω. From (5.5) then it follows that sujmh

+gu∗jhis on ∂Ω1 in components anti-symmetric and symmetric. Beside the boundary symmetry, we canshow even more. Namely for all vh ∈ VΩ1

h ∩H10 (Ω1) and qh ∈ QΩ1

h , we have that(5.6)ah(suj

m

h+ gu∗jh ,vh) + b(vh, s

pj

m

h+ gp∗jh) = 2ah(suj

m

h,vh) + 2b(vh, s

pj

m

h) = 2a(suj ,vh) + 2b(vh, s

pj ),

bΩ1(suj

m

h+ gu∗jh , qh) = 2bΩ1

(sujm

h, qh) = 2bΩ1

(suj , qh).

Noting that the differential operators in polar coordinates preserve the symmetry properties in thefollowing symbolic way (i.e., S stands for a symmetric, A for an anti-symmetric component, seeagain Table 1):

∇(AS

)=

(A SS A

), div

(AS

)= A,

we can conclude from (5.6) that sujmh

+ gu∗jh , spjm

h+ gp∗jh inherit the (anti-)symmetry in Ω1 of the

original functions suj , spj , respectively.

Now, we are ready to formulate that the combination of the singular functions in the schemedoes not contribute to the pollution.

Theorem 13. Let (A1)–(A3) hold, ω < ω3, 1 − λ1 < α < min1,Re(λ3) − 1, and let (Sh)additionally hold for ω ∈ [ω1, ω3). Then, we a priori have for i, j ∈ 1, 2 such that i 6= j thefollowing:

a(sui − sui

mh , s

uj − suj

m

h

)− dh

(sui

mh , s

ujm

h

)+ 2b

(sui − sui

mh , s

pj − s

pjm

h

)= O(h2).(5.7)


Ω1

suppη

.

Ω1 Ω2

. ∂Ω2 ∂Ω1

0

1

η

χ 1–χ

radius r

Figure 3. Illustration of the h-independent domains Ω1 (red), Ω2 (blue), andelement layer Sh (green). The support (and also domain where η = 1) of thecut-off η is bigger than Ω1. The last plot depicts the cut off function χ and itscomplement 1− χ from origin in radial direction. Also, without loss of generality,we assume the support of Eh to be in Ω1\Ω2.

Proof. First, let us focus on the case when ω ∈ [ω1, ω3). By the modified Galerkin orthogonality ofsui

mh and suj

mh

, we can rewrite the terms on the left-hand side of (5.7) by:

a(sui − suimh , s

uj − suj

m

h)− dh(sui

mh , s

uj

m

h) + 2b(sui − sui

mh , s

pj − s

pj

m

h)

= a(suj − sujm

h, sui )− b(sui

mh , s

pj − s

pj

m

h) + 2b(sui , s

pj − s

pj

m

h)

= a(suj − sujm

h, sui )− b(sui

mh , s

pj ) + b(sui , s

pj

m

h) + 2b(sui , s

pj − s

pj

m

h)

= a(suj − sujm

h, sui ) + b(sui , s

pj − s

pj

m

h) + b(sui − sui

mh , s

pj ).

Since the singular functions are smooth in the part of Ω without some fixed surrounding of thesingular corner, we prove the desired convergence rate separately on two parts of the domainΩ using localization techniques of [NS74]. Hence, let us define the sub-domain Ω2 such thatSh ⊂ Ω2 ⊂ Ω1 ⊂ x ∈ Ω; η(x) 6= 0 ⊂ Ω, see Fig. 3, such that the boundary coincides withthe initial partition of Ω. Also, let us introduce a smooth cut-off function χ(r) (cutting off thesurroundings of the singular point), such that χ ≡ 1 in Ω \ Ω1 and χ ≡ 0 in Ω2, being additionallysymmetric in the θ−direction. Since (χsui , χs

pi ) and also (Iuh (χsui ), Iph(χspi )) vanish on Sh, we have

a(suj − sujm

h, Iuh (χsui )) + b(Iuh (χsui ), spj − s

pj

m

h) = 0,

b(suj − sujm

h, Iph(χspi )) = 0,

and thus,

a(suj − sujm

h, sui ) + b(sui , s

pj − s

pj

m

h) + b(sui − sui

mh , s

pj )

= a(suj − sujm

h, χsui ) + b(χsui , s

pj − s

pj

m

h) + b(sui − sui

mh , χs

pj )

+ a(suj − sujm

h, (1− χ)sui ) + b((1− χ)sui , s

pj − s

pj

m

h) + b(sui − sui

mh , (1− χ)spj )

=a(suj − suj

m

h, χsui − Iuh (χsui )) + b(χsui − Iuh (χsui ), spj − s

pj

m

h) + b(sui − sui

mh , χs

pj − I

ph(χspj ))

1

+a(suj − suj

m

h, (1− χ)sui ) + b((1− χ)sui , s

pj − s

pj

m

h) + b(sui − sui

mh , (1− χ)spj )

2.

Adopting the local error estimates derived in [AL95, Theorem 5.3, and Sec.6] for the modifiedscheme, we obtain that

‖suj − sujm

h‖1,Ω\Ω2

+ ‖spj − spj

m

h‖0,Ω\Ω2

. h(‖suj ‖2,Ω\Ω3+ ‖spj‖1,Ω\Ω3

),

where Ω3 ( Ω2, again with boundary coinciding with the initial partition of Ω. Hence, using thesmoothness of the singular functions on Ω \Ω3 and the interpolation estimates (3.1), we obtain the


bound for the first bracket, i.e.:. . .

1.(‖suj − suj

m

h‖1,Ω\Ω2

+ ‖spj − spj

m

h‖0,Ω\Ω2

) (‖sui − Iuhsui ‖1,Ω\Ω3

+ ‖spj − Iphspj‖0,Ω\Ω3

). h2(‖suj ‖2,Ω\Ω3

+ ‖spj‖1,Ω\Ω3).

Let us then study the terms of . . . 2. By the mutual orthogonality of the singular functions,we obtain that a(sui , (1 − χ)suj ) = 0, the approximations, however, do not need to be mutuallyorthogonal due to the possible non-symmetry of the global mesh, as we stated at the beginning ofthis section. To overcome this property, we rewrite the terms of . . . 2 by the local, symmetrizedapproximations gu∗jh + suj

mh

. Namely by the symmetry argument from Lemma 12, it follows that

1

2

(ah(gu∗jh + suj

m

h, (1− χ)sui ) + b((1− χ)sui , g

p∗jh + spj

m

h) + b(gu∗jh + suj

m

h, (1− χ)spi )

)= 0,

hence, adding it to . . . 2, we obtain for β = 1− λ1 + ε, ε sufficiently small such that β < 1/2:

. . .

2=

1

2

[a(gu∗jh − suj

m

h, (1− χ)sui

)+ b((1− χ)sui , g

p∗jh − s

pj

m

h

)+ b(gu∗ih − sui

mh , (1− χ)spj

)]=

1

2

[a(gu∗jh − suj

m

h, (1− χ)sui − Iuh ((1− χ)sui )

)+ b((1− χ)sui − Iuh ((1− χ)sui ), gp∗jh − s

pj

m

h

)+ b(gu∗ih − sui

mh , (1− χ)spj − I

ph((1− χ)spj )

)+ dh

(gu∗jh − suj

m

h, Iuh ((1− χ)sui )

)]. ‖(1− χ)sui − Iuh ((1− χ)sui )‖H1

β(Ω1)

(‖gu∗jh − suj

m

h‖H1−β(Ω1) + ‖gp∗jh − s

pj

m

h‖L2−β(Ω1)

)+ ‖(1− χ)spj − I

ph((1− χ)spj )‖L2

β(Ω1)‖gu∗ih − suimh ‖H1

−β(Ω1)

+ ‖gu∗jh − sujm

h‖H1−β(Ω1)‖Iuh ((1− χ)sui )‖H1

β(Sh).

Since the gradient of the cut-off function χ is independent of h, the terms with interpolants can bebounded by the regularity of particular singular functions and thus

(5.8)‖(1− χ)sui − Iuh ((1− χ)sui )‖H1

β(Ω1) . h‖sui ‖H2β(Ω) . h,

‖(1− χ)spj − Iph((1− χ)spj )‖L2

β(Ω1) . h‖spj‖H1

β(Ω) . h.

Also, the Sh-supported term is of order h, namely we have that r . h on Sh and thus by a scalingargument we have

‖Iuh ((1− χ)sui )‖H1β(Sh) = ‖Iuhsui ‖H1

β(Sh) . hβ‖sui ‖L∞(Sh) . h

βhλi . h.

Let us now investigate the terms in the bracket, i.e., ‖gu∗jh − sujmh‖H1−β(Ω1) + ‖gp∗jh − s

pj

m

h‖L2−β(Ω1).

From the definition (5.1), (gu∗jh − sujmh, gp∗jh − s

pj

m

h) is a solution of the discrete Stokes problem in Ω1

with homogeneous right-hand side and non-homogeneous boundary conditions (5.2) respectively(5.3). First note, that the result (2.3) from [GS06] is generalizable for spaces H1

−β(Ω)2 × L2−β(Ω)

since 1− λ1 < β < 1/2. One can also obtain the a priori estimates for the discrete Stokes problem,due to the discrete inf-sup (3.6), note that by the construction we have gp∗jh − s

pj

m

h∈ L2

0(Ω1), and

the discrete coercivity in weighted spaces of the form a(·, ·), see, e.g., [D’A12, Theorem 3.4]. Thus,together with (A1)–(A3) we have for 1− λ1 < β < 1/2:

‖gu∗jh − sujm

h‖H1−β(Ω1) + ‖gp∗jh − s

pj

m

h‖L2−β(Ω1) . ‖gu∗jh − suj

m

h‖H

1/2−β (∂Ω1)

.(5.9)

Let us now consider a H1−stable extension operator Eh : Vh|∂Ω1→ Vh|Ω1

with a support in

Ω1\Ω2, see Fig 3. Note, that we can find such an extension since suj (as well as sujmh

) are zero on the

boundary of Ω. The special choice of the support of Eh directly gives ‖Eh([gu∗jh−sujmh

]∂Ω1)‖H1−β(Ω1) .

‖Eh([gu∗jh − sujmh

]∂Ω1)‖H1(Ω1). Combining these observations with (5.9), trace theorem, triangle


inequality and the trace estimate (5.4) yields:

‖gu∗jh − sujm

h‖H1−β(Ω1) + ‖gp∗jh − s

pj

m

h‖L2−β(Ω1)

. ‖gu∗jh − sujm

h‖H

1/2−β (∂Ω1)

. ‖Eh([gu∗jh − sujm

h]∂Ω1

)‖H1−β(Ω1)

. ‖Eh([gu∗jh − sujm

h]∂Ω1

)‖H1(Ω1) . ‖gu∗jh − sujm

h‖H

1/200 (∂Ω1\(∂Ω∩∂Ω1))

. ‖suj − sujm

h‖H

1/200 (∂Ω1\(∂Ω∩∂Ω1))

. ‖suj − sujm

h‖H1(Ω\Ω1).

Similarly as above, we can then conclude using the same localization argument and the smoothnessof suj , s

pj away from the singularity that

‖gu∗jh − sujm

h‖H1−β(Ω1) + ‖gp∗jh − s

pj

m

h‖L2−β(Ω1) = O(h),

which together with (5.8) implies that . . . 2 = O(h2) and thus the desired result for the case ofω ∈ [ω1, ω3).

Now, let us study the case for which the local symmetry of the mesh can be dropped, i.e.,ω ∈ (π, ω1). We define a nodal interpolation Iuh with support outside Sh, i.e., Iuhsui (v) = sui (v) for

all vertices which are not contained in Sh and Iuhsui = 0 on Sh. Then dh(sujmh, Iuhsui ) = 0, and we

can rewrite the left-hand side of (5.7) by the modified Galerkin orthogonality as:

a(suj − sujm

h, sui − Iuhsui ) + b(sui − Iuhsui , s

pj − s

pj

m

h) + b(sui − sui

mh , s

pj − I

phspj ).(5.10)

Since ω < ω1, we have that λ1 + λ2 > 2, and thus there exists an exponent γ, such thatH2

1−λj+ε(Ω)2 → H2−γ(Ω)2 and H2

1−λi+ε(Ω)2 → H2γ(Ω)2. Then, (sui , s

pi ) ∈ H2

γ(Ω)2 ×H1γ(Ω) and

(suj , spj ) ∈ H2

−γ(Ω)2 ×H1−γ(Ω). Also, the interpolation error for Iuh can be estimated by the triangle

inequality, (3.1) and a scaling argument since (Iuhsui − Iuhsui ) 6= 0 on O(h)-surrounding of thesingular point, and the fact that hγ . h1−λi , i.e.,

‖∇(sui − Iuhsui )‖0,γ 6 ‖∇(sui − Iuhsui )‖0,γ + ‖∇(Iuhsui − Iuhsui )‖0,γ . h+ hγ‖sui ‖L∞(Sh) 6 h.(5.11)

Note that (suj , spj ) ∈ H2

−γ(Ω)2 × H1−γ(Ω) and thus the estimate for the smooth reminder in

Theorem 11 holds also for (suj , spj ). This, together with (3.1) and (5.11) for the i-th singular

function, gives us by the Cauchy–Schwarz inequality applied on (5.10) the desired estimate (5.7)for ω < ω1.

6. Proof of Theorem 4

In order to obtain the L2α(Ω)2 error estimate as stated in (2.15), we use decomposition from

Lemma 3 on the solution (u, p), i.e.,

u =

2∑i=1

cui sui + U, p =

2∑i=1

cui spi + P, with

2∑i=1

|cui |+ ‖U‖2,−α + ‖P‖1,−α . ‖f‖0,−α.(6.1)

Setting u := u−U and p := p− P , we get by the triangle inequality

‖u− umh ‖0,α 6 ‖u− um

h ‖0,α + ‖U−Umh ‖0,α,

and thus, in view of Theorem 11, it is sufficient to consider only ‖u− umh ‖0,α in more detail. We

start with the dual problem

−∆v +∇z = r2α(u− umh ) in Ω,

div v = 0 in Ω,

v = 0 on ∂Ω.

From the weak formulation of the system above, noting that r2α(u − umh ) ∈ L2

−α(Ω)2, we canidentify by the modified Galerkin orthogonality (2.7) the desired velocity approximation error in


the weighted Lebesgue spaces:(6.2)

‖u− umh ‖20,α = (r2α(u− um

h ), u− umh ) = a(v, u− um

h ) + b(u− umh , z)

= a(u− umh ,v − vm

h )− dh(umh ,v

mh ) + b(u− um

h , z − zmh ) + b(v − vm

h , p− pmh ).

Since r2α(u− umh ) ∈ L2

−α(Ω)2 and ω < ω3, ω 6= ω2, we use the decomposition from Lemma 3 alsoon the solution of the dual problem (v, z):

v =

2∑i=1

cvi sui + V, z =

2∑i=1

cvi spi + Z, with

2∑i=1

|cvi |+ ‖V‖2,−α + ‖Z‖1,−α . ‖u− umh ‖0,α.(6.3)

And thus, due to the linearity of the system, we can rewrite the error (6.2) as

‖u− umh ‖20,α

= a( 2∑i=1

cui (sui − suimh ),

2∑j=1

cvj (suj − suj

m

h) + (V −Vm

h ))− dh

( 2∑i=1

cui sui

mh ,

2∑j=1

cvj suj

m

h+ Vm

h

)

+ b( 2∑i=1

cui (sui − suimh ),

2∑j=1

cvj (spj − s

pj

m

h) + (Z − Zm

h ))

(6.4)

+ b( 2∑i=1

cvi (sui − sui

mh ) + (V −Vm

h ),

2∑j=1

cuj (spj − spj

m

h)).

The bilinearity of the particular functionals and simple rearrangement of the terms yields

‖u− umh ‖20,α

=

2∑i,j=1

1

2(cui c

vj + cuj c

vi )[a(sui − sui

mh , s

uj − suj

m

h

)− dh

(sui

mh , s

uj

m

h

)+ 2b

(sui − sui

mh , s

pj − s

pj

m

h

)]1

+

2∑i=1

cui[a(sui − sui

mh ,V −Vm

h

)−dh

(sui

mh ,V

mh

)+b(sui −sui

mh , Z−Z

mh

)+b(V−Vm

h , spi −s

pi

mh

)]2.

By the modified Galerkin orthogonality (2.7) for V −Vmh , we can rewrite the terms of the second

sum by the help of the interpolations Iuhsui (see definition at the end of the proof of Theorem 13)and Iphs

pi , and thus, by the Cauchy–Schwarz inequality, interpolation estimates (5.11) with γ = α

and (3.1), Theorem 11, (6.1), and (6.3), we obtain:

2∑i=1

...

2=

2∑i=1

cui

[a(sui − Iuhsui ,V −Vm

h

)+ b(sui − Iuhsui , Z − Zm

h

)+ b(V −Vm

h , spi − I

phspi

)].

2∑i=1

|cui |(‖∇(sui − Iuhsui )‖0,α + ‖spi − I

phspi ‖0,α

)(‖V −Vm

h ‖1,−α + ‖Z − Zmh ‖0,−α)(6.5)

. h2 ‖f‖0,−α‖u− umh ‖0,α.

Let us turn our attention to the fully mixed terms of . . . 1. By the triangle inequality, anexplicit integration of the singular functions on Sh and their interpolation errors from Theorem 7,and the Cauchy–Schwarz inequality, we obtain

gh(sui , spi ) . ‖s

ui − sui

mh ‖

21 + ‖∇sui ‖2L2(Sh) + 2‖sui − sui

mh ‖1‖s

pi − s

pi

mh ‖0 . h

2λi , i = 1, 2,(6.6)

and thus, gh(su2 , sp2) . h2 for ω < ω2. This is reflected in the assumption of the theorem in the way

that for ω < ω2, we control the pollution of the first singular function only. Hence, the fully mixed


κ 1

h

(κ1+1)h

(κ1+

κ 2)h

(κ1+

κ 2+

1)h

Figure 4. Graphical representation of S1h (first element ring–red) and S2

h (secondelement ring–blue) domains in a L-shape domain, including the their bounds withrespect to κ1 and κ2.

terms of . . . 1 are easily estimated by the assumption, (6.6), Theorem 13, and (6.1):

2∑i=1

...

1. h2 ‖f‖0,−α‖u− um

h ‖0,α.

All together, dividing (6.4) by ‖u− umh ‖0,α, we obtain the required L2

α(Ω) error estimate. Thesecond error estimate in (2.15) can be derived in a similar fashion as in Theorem 11, this means,splitting the velocity error into the approximation and interpolation error and using (3.1), (3.2) andthe already proven L2

α(Ω) bound in (2.15). For the pressure similarly, whereas the approximationerror is bounded by the discrete inf-sup condition (3.6). By this, we have finished the proof ofTheorem 4.

7. Construction of the modification dh(·, ·)

In the previous sections, we have formulated an abstract condition (2.14) for which optimalorder convergence holds, however, this does not a priori imply the existence of a modificationdh(·, ·) for which (2.14) is satisfied. This will be the subject of this section. We concentrate on atwo parameter modification for ω ∈ (ω2, ω3) and comment later on how to obtain a one parametermodification in the case of ω ∈ (π, ω2).

We closely follow the ideas of [ERW14] and define

(7.1)

dh(vh,wh) := γ1d1h(vh,wh) + γ2d

2h(vh,wh)

= γ1

ˆS1h

∇vh : ∇wh dx+ γ2

ˆS2h

∇vh : ∇wh dx,

where S1h and S2

h are mutually disjoint element layers around the singular point with maximumwidths (κ1 + 1)h and (κ2 + 1)h, namely

S1h := int

(∪T∈ThT ; dist(0, T ) 6 κ1h

), S2

h := int(∪T∈ThT ; κ1h < dist(0, T ) 6 (κ2 + κ1)h

),

thus, from (2.6), Sh = int(S1h ∪ S2

h) and κ = κ1 + κ2. The specification of κ1 and κ2 will be givenin the proof of Theorem 15. For each 0 < R < 1 and each γ ∈ B∞R (0), BpR(0) := (γ1, γ2) ∈R2 : ‖(γ1, γ2)‖lp 6 R, 1 ≤ p ≤ ∞, it is obvious that the assumptions (A1)–(A3) hold. Sincethe modified Galerkin approximation of (sui , s

pi ) depends on the parameter γ = (γ1, γ2)>, we will

denote it from now by1 (R(γ)sui , R(γ)spi ) ∈ Vh ×Qh. In order to satisfy (2.14), we construct, atfirst, a sequence γhh>0 for which the energy defect function will be zero for each h > 0. More

1For the sake of simpler notation, we denote the operator R by the same letter for both approximations, for

velocity and pressure variable.


precisely, we look for γh = (γh,1, γh,2)> such that

gh(γh) =(gh,1(γh), gh,2(γh)

)>:=(gh(su1 , s

p1), gh(su2 , s

p2))>

= 0,(7.2)

where gh(sui , spi ) is defined in (2.8). In the rest of the section, we shall denote by γh a possible root

of gh from (7.2), while by γ or γ we denote general parameters from B∞R (0). The approximationsR(γ)sui , R(γ)spi then satisfy by the modified Galerkin orthogonality the following equations:

a(R(γ)sui ,vh)− γ1d1h(R(γ)sui ,vh)− γ2d

2h(R(γ)sui ,vh) + b(vh, R(γ)spi ) = a(sui ,vh) + b(vh, s

pi ),(7.3)

b(R(γ)sui , qh) = b(sui , qh),(7.4)

for all vh ∈ Vh and qh ∈ Qh, and thus, the properties of ah(·, ·) and b(·, ·) for γ ∈ B∞R (0) yield theh-uniform a priori bounds:

‖R(γ)sui ‖1 + ‖R(γ)spi ‖0 . ‖sui ‖1 + ‖spi ‖0.(7.5)

Before we start with the explicit construction of γhh>0, let us provide an important property ofthe defect function gh.

Lemma 14. The mapping γ onto gh is Lipschitz continuous on B∞R (0), and its gradient withrespect to γ is given by:

∇γgh(γ) =

(−d1

h(R(γ)su1 , R(γ)su1 ) −d2h(R(γ)su1 , R(γ)su1 )

−d1h(R(γ)su2 , R(γ)su2 ) −d2

h(R(γ)su2 , R(γ)su2 )

).(7.6)

Moreover, the djh(R(γ)sui , R(γ)sui

)terms can be bounded for a sufficiently large κ1 > 1 and κ2 = τκ1

with τ ≥ τ0, τ0 > 1 large enough, by:

m(κjh)2λi 6 djh(R(γ)sui , R(γ)sui

)6M(κjh)2λi , i, j = 1, 2.(7.7)

Proof. From the definition (7.3)–(7.4) with γ and test functions R(γ)sui and R(γ)spi , respectively,we have for i = 1, 2:

a(sui , R(γ)sui ) + b(R(γ)sui , spi ) + b(sui , R(γ)spi ) = a(R(γ)sui , R(γ)sui )− γ1d

1h(R(γ)sui , R(γ)sui )

− γ2d2h(R(γ)sui , R(γ)sui ) + b(R(γ)sui , R(γ)spi ) + b(R(γ)sui , R(γ)spi ),

and similarly with γ and test functions R(γ)sui and R(γ)spi :

a(sui , R(γ)sui ) + b(R(γ)sui , spi ) + b(sui , R(γ)spi ) = a(R(γ)sui , R(γ)sui )− γ1d

1h(R(γ)sui , R(γ)sui )

− γ2d2h(R(γ)sui , R(γ)sui ) + b(R(γ)sui , R(γ)spi ) + b(R(γ)sui , R(γ)spi ).

Thus in terms of the definition of gh,i, we obtain for each component the difference:

gh,i(γ)− gh,i(γ) = (γ1 − γ1)d1h(R(γ)sui , R(γ)sui ) + (γ2 − γ2)d2

h(R(γ)sui , R(γ)sui ).(7.8)

Using this and (7.5), the Lipschitz continuity of gh on B∞R (0) follows. Moreover we find for thepartial derivatives:

∂gh,i(γ)

∂γ1= limγ1→γ1

gh,i(γ1, γ2)− gh,i(γ1, γ2)

γ1 − γ1= −d1

h(R(γ)sui , R(γ)sui ),

∂gh,i(γ)

∂γ2= limγ2→γ2

gh,i(γ1, γ2)− gh,i(γ1, γ2)

γ2 − γ2= −d2

h(R(γ)sui , R(γ)spi ),

which in the matrix representation gives (7.6).As last, we show (7.7). Starting with Young inequality (in the form of 2ab 6 2a2 + 1

2b2) and

Theorem 7, we get together with the triangle inequality:

1

2djh(sui , s

ui

)− ch2λi ≤ djh

(R(γ)sui , R(γ)sui

)6

3

2djh(sui , s

ui

)+ 3ch2λi .(7.9)


A straightforward integration of the singular functions on a ball, for ω ∈ (ω2, ω3), with fixedpositive κ and h small enough yieldsˆ

B2κh(0)

∇sui : ∇sui dx = ci(κh)2λi ,(7.10)

and thus, in terms of (7.9) and (7.10), we get the following bounds in (7.9) from the fact thatB2κ1h

(0) ⊂ S1h ⊂ B2

(κ1+1)h(0) and B2(κ1+κ2)h(0) \ B2

(κ1+1)h(0) ⊂ S2h ⊂ B2

(κ1+κ2+1)h(0), see Fig. 4,:

1

2ciκ

2λi1 − c 6 h−2λid1

h


)6

3

2ci(κ1 + 1)2λi + 3c,(7.11)

1

2ci

((κ1 + κ2)2λi − (κ1 + 1)2λi

)− c 6 h−2λid2

h


)6

3

2ci(κ1 + κ2 + 1)2λi + 3c.(7.12)

Setting κ1 > 1 sufficiently large and κ2 := τκ1 with τ as in the assumption of Lemma 14, we cansimplify (7.11)–(7.12) to (7.7).

By means of these properties of the defect function gh, we step to the proof that there actuallyexists a unique sequence γhh>0 fulfilling (7.2).

Theorem 15. For sufficiently large κ1, κ2 ≥ 1 satisfying the relationship from Lemma 14, and foreach sufficiently small h, there exists exactly one γh = (γh,1, γh,2)> ∈ B2

R(0) such that

(7.13) gh(γh) = 0.

Proof. We prove the existence and the uniqueness of γh by the Banach fixed point theorem. Forthat we define a mapping Fh : B∞R (0)→ R2 by:

Fh(γ) :=

(σh−2λ1 gh,1(σ1γ1, σ2γ2) + γ1

σh−2λ2 gh,2(σ1γ1, σ2γ2) + γ2

),

where 0 < σ, and σ1, σ2 ∈ (0, 1]. As we will see below, we will need to bound the gradient ofFh and the mapping itself, and therefore Fh is also scaled in the arguments of gh,i. Due to thedifferentiability of gh(γ), Fh(γ) is also differentiable, hence, together with gh,i(0) . h2λi , seeTheorem 7 for singular functions, we get for all γ, γ ∈ B2

R(0) ⊂ B∞R (0) that

‖Fh(γ)− Fh(γ)‖l2 6 maxξ∈B2

R(0)‖∇γFh(ξ)‖l2 ‖γ − γ‖l2 ,

‖Fh(γ)‖l2 6 ‖Fh(γ)− Fh(0)‖l2 + ‖Fh(0)‖l2 6(

maxξ∈B2

R(0)‖∇γFh(ξ)‖l2 + cσ

)R,

where c is a constant independent of h, κ1, κ2, σ and γ but does depend on R. With respect tothese bounds, if we show that

χ := maxξ∈B2

R(0)‖∇γFh(ξ)‖l2 + cσ 6 1,(7.14)

for some positive σ, then Fh is a contraction on a closed ball, from which follows the existence anduniqueness of γh ∈ B2

R(0) fulfilling (7.13), and thus (7.2). To prove (7.14), we proceed in threeseparate steps.

Step 1: The bound of the gradient of Fh in the spectral norm by the means of the eigenvalues.For each ξ = (ξ1, ξ2)>, we denote the scaled parameter by ζ = (ζ1, ζ2)> := (σ1ξ1, σ2ξ2)> and theprincipal part of ∇γFh(ξ) by B(ξ) := σ−1(Id−∇γFh(ξ)), for simplicity skipping the h argument.Then, by Lemma 14 we have that

B(ξ) =

(σ1h

−2λ1d1h

(R(ζ)su1 , R(ζ)su1

)σ2h

−2λ1d2h


)σ1h

−2λ2d1h


)σ2h

−2λ2d2h


)) .Assuming positive-definitness of B(ξ) +B(ξ)>, we can rewrite the spectral norm of the gradient as:

‖∇γFh(ξ)‖2l2 = Λξmax

(∇γFh(ξ)>∇γFh(ξ)

)6 1− σΛξmin

(B(ξ) +B(ξ)>

)+ σ2Λξmax

(B(ξ)>B(ξ)

),

where Λξmin and Λξmax represent the minimal and maximal eigenvalue of the matrix argument,respectively.


Step 2: Matrix B(ξ) +B(ξ)> is positive definite. Setting σ1 = κ2λ11 κ−2λ2

1 and σ2 = κ2λ11 κ−2λ1

2 ,and thus having strictly σ1, σ2 6 1, we find for τ0 large enough by an elementary computationusing (7.7) and λ2 > λ1, that

det(B(ξ) +B(ξ)>) > 4σ1σ2m2κ2λ1

1 κ2λ22 −M2

(σ2κ

2λ12 + σ1κ

2λ21

)2

> 4κ4λ11

(m2κ

2(λ1−λ2)1 κ

2(λ2−λ1)2 −M2

)= 4κ4λ1

1 (τ2(λ2−λ1)m2 −M2)

> 2κ4λ11 τ2(λ2−λ1)m2 > 0,

and thus, noting that the trace is also positive, we find that B(ξ)+B(ξ)> is strictly positive-definite.Additionally, we can estimate the trace, again using (7.7), the scaling property between κ1 and κ2,and λ2 > λ1, by:

tr(B(ξ) +B(ξ)>) 6 2Mσ1κ2λ11 + 2Mσ2κ

2λ22 6 2Mκ2λ1

1

(κ−2(λ2−λ1)1 + κ

2(λ2−λ1)2

)6 4Mκ2λ2

1 τ2(λ2−λ1).

This means, using the fact that the product of the two eigevalues of B(ξ)+B(ξ)> is the determinantand the sum is equal to the trace, an elementary calculation shows that the minimal eigenvalues

Λξmin of B(ξ) +B(ξ)> are uniformly in ξ bounded from below by

Λξmin >det(B(ξ) +B(ξ)>)

tr(B(ξ) +B(ξ)>)> κ4λ1−2λ2

1

m2

2M.

Step 3: Bound of χ. Denoting by Λ(κ1, τ) an upper bound for the maximal eigenvalue ofB(ξ)>B(ξ) for all ξ ∈ B2

R(0), i.e, Λ(κ1, τ) := maxξ∈B2R(0) Λξmax(B(ξ)>B(ξ)), we can bound χ for

κ1 large enough and σ > 0 small enough by

χ 6

√1− σκ4λ1−2λ2

1

m2

2M+ Λ(κ1, τ)σ2 + cσ 6 1−

(κ4λ1−2λ2

1

m2

4M− c)σ +

Λ(κ1, τ)

2σ2 < 1,

where we used the algebraic inequality√

1 + x 6 1 + 12x. More precisely, κ1 has to be selected

large enough, such that κ4λ1−2λ21 > 4Mm−2c, and σ > 0 small enough that the linear term term

dominates the quadratic one. We recall that 2λ1 − λ2 > 0 for all ω ∈ (π, 2π).

In theory, it is then sufficient to define dh(·, ·) in (7.1) with γh = (γh,1, γh,2)>, where γhsolves (7.2). In practice this would require to change the finite element formulation at eachcomputational level. It is clear, that such a modification of the original scheme would not be of agreat interest. However, the condition (7.2) is over-restrictive. We rather require that the defectfunction gh = O(h2), see the assumptions (2.14) of Theorem 4. This means that we can relax thecondition on the choice of the correction parameter.

As discussed in [ERW14] and [RWW14] for the modified Galerkin approximation or in [ZG78]for the modified finite difference method for Poisson equation, the roots converge to a uniqueasymptotic value with a sufficiently high rate, namely, for an one-correction modification withh2(1−λ1) rate. Adopting the proof for a two-correction modification, the existence of the convergencefollows by the same arguments as in [RWW14], nevertheless, we obtain only h2(1−λ2) convergencefor each component of γh. However, such observation is not sufficient, as we will see later. Yet, ifwe fix the second component of γh to some arbitrary (but admissible) value, the modification willagain adopt the convergence properties of the single-correction method (for the price of a reducedconvergence of g2,h), and thus, a higher order convergence in γ1,h. This motivated our proof of theoptimal convergence of the pollution function gh as stated in the next lemma.

Lemma 16. Let γhh>0 be the sequence of correction parameters solving (7.2) and let dh(·, ·) bedefined for γ? = (γ?1 , γ

?2), γ? = (γ?1 , γ

?2 ) ∈ B2

R(0) such that

|γ?1 − γ1,h| = O(h2(1−λ1)),


where γ1,hh>0 is the unique sequence of correction parameters solving gh,1(γ1,h, γ?2) = 0. Then

for the first component of the pollution function gh, defined in (7.2), it holds:

gh,1(γ?) = O(h2).(7.15)

If additionally

‖γ? − γh‖l2 = O(h2(1−λ2)),(7.16)

then γ? = γ? and the second component of the pollution function gh also converges:

gh,2(γ?) = O(h2).(7.17)

Proof. In the case of the first assumption, i.e., assuming a correction with γ? = (γ?1 , γ?2 ), we directly

have by Lipschitz continuity of gh for each γ ∈ BR(0), (7.8) with Young inequality:

(7.18)|gh,1(γ?)| = |gh,1(γ?)− gh,1(γ1,h, γ

?2)|

. |γ?1 − γ1,h|(‖∇R(γ?)su1‖2L2(Sh) + ‖∇R(γ1,h, γ

?2)su1‖2L2(Sh)

)= O(h2).

Note, that the uniqueness of γ1,hh>0 ∈ [0, R] follows by a suitable adaptation of Theorem 15. Toprove the second assertion (7.17), we show first that γ?1 = γ?1 by contradiction. Let us thus assumethat γ?1 6= γ?1 , then we have for a suitable ξ ∈ [0, R]:

gh,1(γ?1 , γ?2 ) = gh,1(γ?1 , γ

?2) + (γ?1 − γ?1 )

∂gh,1∂γ1

(ξ, γ?2).

By (7.6) and (7.7), we know that∂gh,1∂γ1

(ξ, γ?2 ) > ch2λ1 , and thus, together with (7.18):

gh,1(γ?1 , γ?2 ) > ch2λ1 .(7.19)

At the same time, by assumption (7.16) and the same arguments as for (7.18) we obtain:

gh,1(γ?1 , γ?2) 6 c‖γ? − γh‖l2 h2λ1 6 ch2λ1+2(1−λ2),

which is, due to λ2 < 1 for all ω ∈ (ω2, ω3), a contradiction to (7.19), and thus γ?1 = γ?1 , togetherwith gh,1(γ?) = O(h2). The quadratic convergence of the second component gh,2(γ?) then followslikewise as for (7.18).

Remark 17. The situation of ω < ω2 is much simpler. We define dh(·, ·) in (7.1) by

dh(vh,wh) := γ1d1h(vh,wh) = γ1

ˆS1h

∇vh : ∇wh dx,(7.20)

and thus the formulation of γh reduces to the scalar one. Consequently, Lemma 15 can be simplifiedand in Lemma 16, γ?2 is automatically zero and γ1,h = γ1,h as well as γ?1 = γ?1 , and thus we havedirectly for the pollution function: gh = g1,h = O(h2).

By the existence of the asymptotic γ? and the convergence result (7.15) and (7.16), one can usein (7.1), respectively (7.20), the correction parameters γ1 = γ?1 and γ2 = γ?2 . This means, knowingγ?, the modification of the standard finite element scheme is simplified to a single change of afew entries of the linear system. This strategy is used in the following section for the numericalcomputations.

8. Numerical results

In the previous sections, we have proven approximation properties of the modified Galerkinscheme for the prototypical P1–isoP2/P1 discretization. However, to manifest the flexibility ofthe method, we, additionally to P1–isoP2/P1 element, consider further stable and stabilized finiteelement discretizations, namely the Mini element, the Taylor–Hood element, the P2–P0 elementand the stabilized P1–P1 element. For that, we introduce the following finite element spaces:

Vkh =

vh ∈ H1

0 (Ω)2 : vh|T ∈ Pk(T ) ∀T ∈ Th,

Qkh =qh ∈ L2

0(Ω) ∩ C(Ω) : qh|T ∈ Pk(T ) ∀T ∈ Th,


for k > 1, the discontinuous pressure space:

Q0h =

qh ∈ L2

0(Ω) : qh|T ∈ P0(T ) ∀T ∈ Th,

and the space of element-wise bubble functions:

Bh =vh ∈ C(Ω) : vh|T ∈ spanλ1,Tλ2,Tλ3,T ∀T ∈ Th

2,

where λi,T , i = 1, . . . , 3 denote the barycentric coordinates on the element T , see also [ABF84].The Mini element is then defined via the product V1

h⊕Bh×Q1h, the Taylor–Hood element is given

by V2h ×Q1

h, the P2–P0 element by V2h ×Q0

h and the P1–isoP2/P1 element by V1h/2 ×Q

1h.

The energy corrected variational formulation for the stabilized P1–P1 element is then formulatedas: Find (um

h , pmh ) ∈ Vh ×Qh = V1

h ×Q1h such that

ah(umh ,vh) + b(vh, p

mh ) = 〈f ,vh〉 ∀vh ∈ Vh,

b(umh , qh)− ch(qh, p

mh ) = 〈g, qh〉 ∀qh ∈ Qh,

where the additional stabilization terms are of the form

ch(qh, ph) :=∑T∈Th

δT h2T

ˆT

∇ph · ∇qh dx, 〈g, qh〉 := −∑T∈Th

δT h2T

ˆT

f · ∇qh dx,

with a stabilization parameter δT > 0 such that

δT =1

12

(1− γi)−1 for T ∈ Sih, i = 1, 2,

1 else.

This stabilization in the case of standard finite element method is well understood and known as thePSPG- or Brezzi–Pitkaranta stabilization, see, e.g., [BD88]. The generalization of the stabilizationto our modified scheme can be found in [HJP+15], where also the equivalence to the Mini elementis derived.

Unless not further specified, we shall always consider a computational setting for which (su1 +su2 , s

p1 + sp2 − |Ω|−1〈sp1 + sp2, 1〉Ω) is the exact solution, were the cut-off function η is set to be equal

one in Ω. This implies the right-hand side f = 0 and a non-trivial Dirichlet datum g.

8.1. Example: L-shape domain for different discretizations. At first, we consider thestandard L-shape domain Ω = (−1, 1)2\([0, 1] × [−1, 0]) with the interior angle ω = 3/2π. Thetriangulation of the domain Ω is depicted in Fig. 5, including the first and second element rings S1

h

and S2h.

Figure 5. Symmetric mesh around the singular corner and correction patches,S1h (red) and S2

h (blue), for two different refinements.

In the following, we present the convergence rates of the error for the energy corrected finiteelement method, using different types of discretizations. As it was pointed out previously, weexpect at least second and first order of convergence in weighted L2

α(Ω) norms, for velocityand pressure, respectively, c.f. also estimate (2.15). The corresponding asymptotic correctionparameters (γ?1 , γ

?2)> for the mesh as depicted in Fig. 5 and different discretization methods are

presented in Tab. 2. We recall that the local mesh influences the value of γ?.The numerical results for the velocity are presented in Fig. 6(a). As expected, we observe

second order of convergence in L2αu(Ω), with αu = 1− λ1 and λ1 = 0.54448... for all considered


Discretization γ? = (γ?1 , γ?2)>

Mini (0.24010,−0.07106)>

P2–P1 (−0.18050, 0.06703)>

P2–P0 (0.15777,−0.15030)>

PSPG (0.16171,−0.18398)>

P1–isoP2/P1 (−0.27637, 0.14510)>

Table 2. Asymptotic correction parameters γ? for different discretizations.

102 103 104 105

10−5

10−4

10−3

10−2

10−1

DoFs

erro

r

MiniP2–P1

P2–P0

PSPGP1–iso

O(h2)

(a) ‖u− umh ‖0,αu .

102 103 104 105

10−3

10−2

10−1

100

DoFs

erro

r

MiniP2–P1

P2–P0

PSPGP1–iso

O(h3/2)

(b) ‖p− pmh ‖0,αp .

Figure 6. Order of convergence for velocity and pressure for different discretizations.

discretizations. We note that for Taylor–Hood elements a higher convergence rate can be recoveredby increasing the weight αu, which will be discussed in more detail in Sec. 8.3. The correspondingresults for the pressure are presented in Fig. 6(b). The weight for the pressure is chosen slightlyhigher, namely to be αp = 3/2− λ1. Due to this increased weight in the pressure norm, we canobtain super convergence effects. For regular situations, we refer to [ETX11]. This means, aconvergence rate of 3/2, in the case of the Mini, PSPG and P1–isoP2/P1 discretization. For theP2–P0 element we observe linear convergence, as expected also by the best approximation property.

8.2. Example: L-shape domain – convergence for the modified and standard scheme.In order to demonstrate the efficiency gain of the energy corrected finite element method, we alsopresent the above results in more detail for one particular discretization. For simplicity, we studyhere the case of the Mini element, other descritizations perform in a similar way. As before wechoose αu = 1− λ1 and αp = 3/2− λ1.

In Tab. 3, we present the errors and their convergence rates for the uncorrected and correcteddiscretization, both in standard L2(Ω) and weighted L2

α(Ω) norms. For comparison, the asymptoticconvergence rate of the L2(Ω) error of the standard approximation is for the velocity 2λ1 and forthe pressure λ1, where λ1 = 0.54448.... In the case of the standard finite element method, we alsoobserve, that a simple change of the considered norm in which we measure the error has a minorinfluence on the magnitude, whereas the rate of convergence remains unaltered. On the other hand,the modified scheme exhibit an optimal convergence rate (for linear approximations) of order 2 and3/2 for velocity and pressure, respectively, if the correction parameter γ is close enough to optimal.

In Tab. 4, we present the convergence rates and the errors for two different guesses of γ. As onecan see, the correction parameter has to be chosen close enough to the asymptotic one in order torecover the optimal convergence rates.

For completeness, we also present the errors and corresponding convergence rates for the defectfunction gh(γ) in Tab. 5. As it was shown in Theorem 4, the energy defect function has to be of


γ = 0 γ? = (0.24010,−0.07106)>

‖u− uh‖0 eoc ‖p− ph‖0 eoc ‖u− umh ‖0 eoc ‖p− pm

h ‖0 eoc

1.33386e–01 – 3.70246e+00 – 1.31908e–01 – 2.74998e+00 –4.68432e–02 1.51 1.00100e+00 1.89 4.40878e–02 1.58 6.93021e–01 1.991.93210e–02 1.28 4.85869e–01 1.04 1.49193e–02 1.56 3.96491e–01 0.818.26076e–03 1.23 2.74084e–01 0.83 5.00124e–03 1.58 2.47925e–01 0.683.61726e–03 1.19 1.68209e–01 0.70 1.69606e–03 1.56 1.60596e–01 0.631.61404e–03 1.16 1.08309e–01 0.64 5.78727e–04 1.55 1.06215e–01 0.607.31498e–04 1.14 7.16586e–02 0.60 1.97968e–04 1.55 7.12092e–02 0.58

‖u− uh‖0,αu eoc ‖p− ph‖0,αp eoc ‖u− umh ‖0,αu eoc ‖p− pm

h ‖0,αp eoc

1.05328e–01 – 2.88733e+00 – 9.91323e–02 – 2.03148e+00 –3.16034e–02 1.74 6.35434e–01 2.18 2.61978e–02 1.92 3.36558e–01 2.591.17120e–02 1.43 2.30700e–01 1.46 6.63126e–03 1.98 1.18968e–01 1.504.74319e–03 1.30 9.07135e–02 1.35 1.62502e–03 2.03 4.24418e–02 1.492.04906e–03 1.21 3.75757e–02 1.27 4.03809e–04 2.01 1.50078e–02 1.509.17095e–04 1.16 1.62799e–02 1.21 1.01021e–04 2.00 5.25528e–03 1.514.18764e–04 1.13 7.29422e–03 1.16 2.51859e–05 2.00 1.85082e–03 1.51

Table 3. Comparison of the discretizations in case of the Mini element. Wepresent the error and its convergence rate for modified and standard Galerkinapproximation, as well as a comparison with respect to the weight of the L2(Ω)norm.

γ = (0.2,−0.1)> γ = (0.23,−0.08)>

‖u− umh ‖0,αu eoc ‖p− pm

h ‖0,αp eoc ‖u− umh ‖0,αu eoc ‖p− pm

h ‖0,αp eoc

1.04045e–01 – 2.18207e+00 – 2.84376e–02 – 3.49635e–01 –2.85719e–02 1.86 3.90083e–01 2.48 7.55233e–03 1.91 1.26219e–01 1.477.82421e–03 1.87 1.50969e–01 1.37 2.01295e–03 1.91 4.54561e–02 1.472.48892e–03 1.65 5.78521e–02 1.38 5.81264e–04 1.79 1.62093e–02 1.499.75955e–04 1.35 2.23724e–02 1.37 1.85978e–04 1.64 5.76718e–03 1.494.22485e–04 1.21 8.92966e–03 1.33 6.66982e–05 1.45 2.08878e–03 1.471.90103e–04 1.15 3.72638e–03 1.26 2.65056e–05 1.33 7.74483e–04 1.43

Table 4. Errors and convergence rates for the Mini element, for a rough guess of γ.

γ = 0 γ? = (0.24010,−0.07106)> γ = (0.2,−0.1)>

|g1(γ)| eoc |g2(γ)| eoc |g1(γ)| eoc |g2(γ)| eoc |g1(γ)| eoc |g2(γ)| eoc

9.45e–01 – 8.30e–04 – 2.67e–01 – 1.69e–02 – 4.32e–01 – 1.05e–02 –2.02e–01 2.22 6.89e–04 0.27 5.98e–03 5.48 9.69e–04 4.12 8.35e–02 2.37 1.76e–03 2.588.46e–02 1.26 2.54e–04 1.44 1.87e–03 1.68 2.53e–04 1.94 3.73e–02 1.16 5.41e–04 1.703.77e–02 1.16 9.17e–05 1.47 5.02e–04 1.90 5.16e–05 2.29 1.68e–02 1.15 1.74e–04 1.631.73e–02 1.12 3.14e–05 1.55 1.04e–04 2.27 9.21e–06 2.49 7.72e–03 1.13 5.49e–05 1.678.06e–03 1.10 1.03e–05 1.61 1.65e–05 2.65 1.23e–06 2.90 3.58e–03 1.11 1.70e–05 1.70

Table 5. Errors and convergence rates for the defect functions for the Mini element.

second order in order to obtain optimal convergence rates. As we can see, in our computationalsetting the convergence rate is even higher. Naturally, we observe a reduced convergence rate ofthe defect function for the standard scheme and in the case of a rough guess, which is the cause ofthe pollution as you can see in Tab. 3 and Tab. 4.


Remark 18. We point out that the correction parameter γ has to be accurate enough in orderto recover the optimal convergence of the method. Also, γ depends on ω, the number of elementsat the singular corner and their shape, but not on the global domain. However, if the correctionparameter is available, the implementation of the energy-correction requires only a rather triviallocal modification in an already existing code.

Figure 7. Initial meshes of the studied angles 8/7π, 5/4π, 3/2π, 7/4π, 15/8π.Note that for ω = 8/7π, we do not require the local symmetry of the mesh.

× × × × ×

π 87π

54π

32π ω3ω2ω1

74π

158 π 2π

0.5

1

2

3

γ? = (γ?1 , γ?2)

>γ? = (γ?1 , 0)>

Figure 8. Convergence rates of ‖u− umh ‖0,αu for the Mini element (×) and the

Taylor–Hood element (), shifted by minus one in order to compare the convergencerates of the errors in L2

α(Ω) norms with the real parts of the eigenvalues λi. Red:αu = αmin = 1− λ1, blue: αu = αmax = min1,Re(λ3)− 1, yellow: αu = 2− λ1.The green line represents λ1 + min2,Re(λ3) − 1.

8.3. Example: Different interior angles ω. The results of this section are graphically presentedin Fig. 8. We sample several interior angles, namely ω ∈ 8/7π, 5/4π, 3/2π, 7/4π, 15/8π, andpresent the convergence rates for the velocity in dependence on the maximal interior angle. Thepressure has a similar behavior, therefore we do not present it here. The convergence rates aregiven in the weighted L2

αu(Ω) norm but shifted by minus one, in order to identify them with theregularities of particular singular functions, hence the eigenvalue-plot is included in the graphic.For ω ∈ (π, ω2), we use a single correction parameter, for ω ∈ (ω2, 2π) we use a modification withtwo correction parameters.


In the theoretical part above, we have always assumed a bound for the weight αu by

αu ∈ [αmin, αmax] := [1− λ1 + ε, min1,Re(λ3)− 1 − ε],(8.1)

securing a simple decomposition of the solution (u, p) into two singular parts and a smoothremainder in H2

−αu(Ω)2 ×H1−αu(Ω). This means, we have always restricted ourselves to the case

when the maximal interior angle was in the range of (π, ω3). With this example, we demonstratehowever, that a further decomposition of the solution with a larger number of considered singularfunctions does not deteriorate the optimal convergence rates of the proposed method, if thecoarse mesh satisfies an additional uniformity at the singular point, see Fig. 7, where the adjoinedelements at the singular point are of the same shape. This means, that the same construction ofthe modification (7.1) recovers the optimal rates even for bigger angles than ω3, see the results forthe Mini element (red cross) in Fig. 8. Note, that the lower bound in (8.1) is essential, while theupper one was only technical, simplifying our error analysis.

The second purpose of this example is to demonstrate an increased convergence rate for ahigher-order element, here we have chosen the popular Taylor–Hood element, with the samecorrection strategy as for linear approximations. Let us briefly explain the obtained convergencebehavior. For the linear approximations, the best approximation is of second order, this means thatindependently of the chosen αu such that αu > αmin, the convergence order will stay quadratic.On the other hand, for the quadratic scheme, a different choice of αu in (8.1) leads, to a differentconvergence order. In what follows, we focus on two extrema, namely, we identify the rates for theweights αmin and αmax. In the graph, these are the rates laying on the constant 1 (red squares) andthose rates laying on the green curve of λ1 + Re(λ3)− 1 (blue squares), respectively. For simplicity,we only sketch the main difference to linear elements, without giving a rigorous proofs. If we take acloser look on the proof of Theorem 4, we have required for linear approximations and for ω < ω3

that ‖V−Vmh ‖0,−α . h2(‖V‖2,−α+‖Z‖1,−α), which then, independently of α ∈ [αmin, αmax], leads

to the second order of convergence of the pollution function gh. For the quadratic approximations,on the other hand, we require for optimal convergence rates that gh = O(h3), under the additionalassumption that (U, P ) ∈ H3

−α+1(Ω)2 × H2−α+1(Ω). Nevertheless, since we correct only up to

two pollution functions, we get only a reduced order σ < 3. Let us now examine σ, particularlyσmin and σmax, i.e., the rates corresponding to the weights αmin and αmax, respectively, moreclosely. Following (6.5), we have by the Cauchy–Schwarz inequality, for some admissible α anda priori estimates (6.1), suited for the higher order case, i.e., for the case when the smooth reminder(U, P ) ∈ H3

−α+1(Ω)2 ×H2−α+1(Ω):

(8.2)‖u− um

h ‖20,αmin. |cui |‖V −Vm

h ‖0,−α‖sui ‖2,α + · · · . hσmin (‖V‖3,−αmin+1 + ‖Z‖2,−αmin+1) + . . .

. hσmin‖u− umh ‖0,αmin

,

similarly for α = αmax. Note, that the estimates above were consequences of (6.1) and Theorem 11,where the estimation of the gradient of the interpolation error was necessary. Namely, for aquadratic interpolation operator Iu,2h we have for each v ∈ H3

−β+1(Ω)2 with β > 0:

‖∇(v − Iu,2h v)‖0,γ . h1+γ+β‖v‖3,−β+1, −β − 1 6 γ 6 −β + 1,

and thus, we obtain:

‖∇(V −Vh)‖0,−α . ‖∇(V − Iu,2h V)‖0,−α . h1−α+αmin‖V‖3,−αmin+1,

‖∇(V −Vh)‖0,−α . ‖∇(V − Iu,2h V)‖0,−α . h1−α+αmax‖V‖3,−αmax+1.

Choosing the weight α as small as possible, i.e., α = αmin, we can then determine the error estimatein L2

−α(Ω) for (8.2) (and for ‖ · ‖0,αmaxequivalently) by:

‖V −Vh‖0,−α . h2 (‖V‖3,−αmin+1 + ‖Z‖2,−αmin+1) ,

‖V −Vh‖0,−α . hλ1+min2,Re(λ3)−ε (‖V‖3,−αmax+1 + ‖Z‖2,−αmax+1) ,

by which we have established the rates σmin = 2 and σmax = λ1 + min2,Re(λ3) − ε.


In the previous paragraph, we have studied the convergence behavior of the Taylor–Hood elementfor the same correction strategy as for the linear elements. Nevertheless, there are limitations of thisapproach preventing the optimal, i.e., cubic, convergence rate. Namely, for optimal convergence,one requires that (u, p) ∈ H3

α(Ω)2 ×H2α(Ω), where α > 2− λ1. This means, that the upper bound

in (8.1) is restricting, causing the suboptimal rates for all ω ∈ (π, 2π), see the green line in Fig. 8.Note, on the other hand, that by this condition, we have a priori enforced the assumption onthe smooth remainder: (U, P ) ∈ H3

−α+1(Ω)2 ×H2−α+1(Ω). The exact derivation of the necessary

condition on the optimal convergence of higher order element-approximation, however, exceeds theframework of this paper, and thus, it is left to a further study. At this point, we only remark thatfor small ω, we can even observe a third order convergence for a one-parameter modification if theweight is further increased, see the yellow square in the figure in the case of ω = 8/7π.

8.4. Convergence of γh. In this section, we present the convergence of the roots γh as assumedin Lemma 16. For the quadratic convergence of both components of the pollution function gh,as necessary condition of the optimal convergence of the modified scheme, we have assumed inLemma 16 a faster convergence if the second correction parameter is fixed, and a slower convergenceof the roots themselves, namely 2(1− λ1) and 2(1− λ2). Such behaviour is depicted in Fig. 9 inthe case of the PSPG approximation and ω = 3/2π, where all three graphs show the numericalvalues of the roots γ1,h, γ1,h and γ2,h, respectively, the extrapolation lines with a given decay rateand the extrapolated limiting values. As one can see, the asymptotical value of a correction with(γ1,h, γ

?2 )> is the same as the asymptotical value of (γ1,h, γ2,h)>, as stated in the lemma.

Remark 19. In all of our numerical tests, we have observed, that γ1,h + γ2,h = O(h2(1−λ1)). Also,that there exist angles ω (also with dependency on the local mesh) for which γ1,h and γ2,h convergethemselves with the higher rate, namely 2(1− λ1). In our settings it was ω = 7/4π.

1 3 5 7 90.16

0.17

0.18

`

γ1,h

γ?1 = γ?1γ?1 + ch2(1−λ1)

1 3 5 7 9

0.05

0.1

0.15

`

γ1,h

γ?1γ?1 + ch2(1−λ2)

1 3 5 7 9

−0.06

−0.12

−0.18

`

γ2,h

γ?2γ?2 + ch2(1−λ2)

Figure 9. Convergence of γh in the case of PSPG discretization for ω = 3/2π.The curved lines represent the extrapolation through the exact roots in relationto the refinement level `, the red lines represent the extrapolated values of γ?1 , γ?1and γ?2 , respectively.

For an illustration, we also include a set of graphs representing the roots of the pollution functiongh for the PSPG approximation on the fourth computational level, for the angles 5/4π, 3/2πand 7/4π, see Fig. 10. Note, that the angle between the zero-isolines of the two componentsof gh decreases with increasing ω. However, a proper rescaling of the pollution function, i.e.(gh,1 + cr(h) gh,2, gh,2)>, cr being a suitable h-dependent constant, allows us to determine the rootsin a numerically stable way.

The last figure in this section, Fig. 11, depicts a fitting function from which one can roughlyestimate the asymptotic correction parameter in the case of a single correction modification, as wasproposed for the Laplace equation in [RWW14]. The plot is here constructed for the Taylor–Hoodelement V2

h ×Q1h, the other discretizations follow analogously. For this study, a fixed number of

elements attached to the singular corner is chosen, namely n = 6, and multiple angles ω ∈ (π, 2π)


(a) ω = 5/4π (b) ω = 3/2π (c) ω = 7/4π

Figure 10. Roots of the pollution function gh. (a) Scalar function g1,h as functionof γ1. (b) and (c) Scaled isolines of g1,h (red) and g2,h (blue). The roots are forall cases depicted with a black point. PSPG, ` = 4.

are considered. Similarly as in [RWW14], we fit the numerical data γ?1(ω) by an ad hoc function:

γ?fit(ω) := σ1(eσ2(π−ω) − 1) + σ3(ω − π) + σ4(ω − π)2 + σ5(ω − π)3,

where the fitting procedure specified the coefficients to

σ1 = 0.038889804903411, σ2 = 2.3147185162477686, σ3 = 0.08578685441949138,

σ4 = −0.0977865017985656, σ5 = 0.013194580802218785.

2π74π

32π

54π

π

−0.3

−0.2

−0.1

0

Figure 11. Asymptotic values γ?1 (•) and the fitting function γ?fit(ω) (blue).On the right: a typical computational domain for the case ω = 29/16π with 6symmetric elements at the singular corner.

8.5. Example: Flow over step. As the last example, we present the classical benchmark of ”flowover a step”, c.f. [ESW05]. This means, we consider Ω = (−1, 5)×(−1, 1)\[−1, 0]2, with a prescribedinflow on the boundary part −1 × [0, 1], realized via the Dirichlet datum u = (4x1(1− x1), 0)>.Moreover, on 5 × [−1, 1] a free outflow is considered, namely (∇u) n− pn = 0, the remainingpart of the boundary is assumed to be a no slip boundary, i.e. u = 0. The right hand-side isf = 0. We shall note that the (in principle not known) exact solution of this model probleminvolves all singular functions (sui , s

pi ). Moreover, it is important to mention that the boundary

condition around the re-entrant corner is of Dirichlet type. For the discretization, we consider theTaylor–Hood element V2

h ×Q1h and weights αu = 1− λ1, αp = 3/2− λ1 with λ1 = 0.54448.... The

asymptotic correction parameter γ? is obtained by solving a corresponding Dirichlet problem.Corresponding numerical results are presented in Tab. 6. Since the exact solution for this

problem is not known, we consider the solution of refinement level ` = 6 as a reference solutionfor the error evaluation, which is calculated for the corrected and uncorrected individually. Theconvergence rates of the errors are of the same order as in the previous L-shape scheme. Notethat the order of convergence on the last level is slightly improved due the choice of the reference


γ = 0 γ? = (−0.17500, 0.06507)>

‖u− uh‖0 eoc ‖p− ph‖0 eoc ‖u− umh ‖0 eoc ‖p− pm

h ‖0 eoc

4.15531e–02 — 5.83349e–01 – 4.08912e–02 – 5.45423e–01 –1.66969e–02 1.32 3.86987e–01 0.59 1.48869e–02 1.46 3.71528e–01 0.556.54274e–03 1.35 2.55566e–01 0.60 5.15675e–03 1.53 2.51700e–01 0.562.57889e–03 1.34 1.67325e–01 0.61 1.76919e–03 1.54 1.68093e–01 0.589.83051e–04 1.39 1.05425e–01 0.67 5.96888e–04 1.57 1.07863e–01 0.643.09388e–04 1.67 5.83149e–02 0.85 1.81945e–04 1.71 6.00990e–02 0.84

‖u− uh‖0,αu eoc ‖p− ph‖0,αp eoc ‖u− umh ‖0,αu eoc ‖p− pm

h ‖0,αp eoc

2.62859e–02 – 3.07839e–01 – 2.34095e–02 – 1.44190e–01 –9.18468e–03 1.52 1.43760e–01 1.10 6.28190e–03 1.90 4.44960e–02 1.703.41619e–03 1.43 6.38446e–02 1.17 1.57475e–03 2.00 1.40378e–02 1.661.36647e–03 1.32 2.77786e–02 1.20 3.88671e–04 2.02 4.54514e–03 1.635.35830e–04 1.35 1.12699e–02 1.30 9.44222e–05 2.04 1.50045e–03 1.601.68875e–04 1.67 3.62477e–03 1.64 2.13247e–05 2.15 5.22527e–04 1.52

Table 6. Errors for the ”flow over step” example using the P2–P1 element, withand without energy correction.

solution. Also, we would like to mention that the asymptotic correction parameter γ? is in thiscase different from the one presented in Tab. 2, which is due to a different mesh around the singularcorner.

Figure 12. Relative difference of uncorrected and corrected velocity.

Additionally, we present plots for the differences of the uncorrected and corrected velocity inFig. 12. We observe that the main difference occurs not only around the singular corner but also inthe region where typically the separation line occurs. The appearing differences are thereby in therange of 0 to 10%. It is further important to mention, that in the remaining part of the domainthese differences are below 1%. Another possibility to present the non-local difference betweenthese two methods is depicted in Fig. 13. There, we compare the difference of the approximationand the reference solution (of the corrected scheme). While the error for the corrected method isstrongly localized at the singular corner, the error of the standard approximation from the referencesolution is still appearing in a large part of the domain. This naturally reflects the influence of thepollution effect.


Figure 13. Illustration of the pollution effect by |umref−um

h | (left) and |uref−uh|(right) on level ` = 4.

Acknowledgements

The financial support by the German Research Foundation (DFG) trough grant WO 671/11-1is gratefully acknowledged.

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