Comput. Methods Appl. Mech. Engrg. Primal interface...

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Primal interface formulation for coupling multiple PDEs:A consistent derivation via the Variational Multiscale method

Timothy J. Truster, Arif Masud ⇑Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 3129E Newmark Civil Engineering Laboratory, MC-250, Urbana,IL 61801-2352, United States

a r t i c l e i n f o

Article history:Received 30 April 2013Received in revised form 1 August 2013Accepted 13 August 2013Available online 17 September 2013

Keywords:Discontinuous GalerkinVariational Multiscale methodInterfacesNitsche methodMultiple PDEsResidual-free bubbles

a b s t r a c t

This paper presents a primal interface formulation that is derived in a systematic mannerfrom a Lagrange multiplier method to provide a consistent framework to couple differentpartial differential equations (PDE) as well as to tie together nonconforming meshes. Thederivation relies crucially on concepts from the Variational Multiscale (VMS) approachwherein an additive multiscale decomposition is applied to the primary solution field.Modeling the fine scales locally at the interface using bubble functions, consistent resid-ual-based terms on the boundary are obtained that are subsequently embedded into thecoarse-scale problem. The resulting stabilized Lagrange multiplier formulation is convertedinto a robust Discontinuous Galerkin (DG) method by employing a discontinuous interpo-lation of the multipliers along the segments of the interface. As a byproduct, analyticalexpressions are derived for the stabilizing terms and weighted numerical flux that reflectthe jump in material properties, governing equation, or element geometry across the inter-face. Also, a procedure is proposed for automatically generating the fine-scale bubble func-tions that is inspired by a performance study of residual-free bubbles for the interfaceproblem. A series of numerical tests confirms the robustness of the method for solvinginterface problems with heterogeneous elements, materials, and/or governing equationsand also highlights the benefit and importance of deriving the flux and stabilization terms.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

The development of robust techniques for modeling interfaces within the finite element method has been an active areaof research in recent years. The broad range of physical applications and associated numerical methods can be grouped intotwo major categories depending upon whether the interface is permitted to intersect element interiors or is restricted tocoincide with element boundaries. Examples of the former, often referred to as embedded or immersed interface methods,have been applied to problems such as fluid–structure interaction, crack propagation, and evolving phase boundaries inchemical reactions. In the latter case, the interface can arise from coupling nonconforming meshes in domain decompositionmethods or from modeling contact between mechanical bodies. Nonetheless, the underlying issue common to each of theseproblems is the (weak) enforcement of the kinematic or continuity conditions on the fields at the interface when discontin-uous functions are employed.

One classical approach for imposing these constraints in the variational setting is the Lagrange multiplier method. As aprominent example, the discontinuous enrichment method [15] was proposed in order to enrich the finite element space

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⇑ Corresponding author. Tel.: +1 (217) 244 2832; fax: +1 (217) 265 8039.E-mail address: [email protected] (A. Masud).

Comput. Methods Appl. Mech. Engrg. 268 (2014) 194–224

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with analytical functions and weakly enforce continuity of the solutions across element boundaries using Lagrange multi-pliers. The main drawback/difficulty with such methods in general is that the presence of complex interface geometryand discontinuous enrichment functions for the primary field renders the determination of stable multiplier interpolationsnontrivial [7]. In order to admit arbitrary interpolations for the mixed primary-multiplier formulation, Barbosa and Hughes[3] proposed Galerkin least-squares (GLS) stabilizing terms involving the Euler–Lagrange residual on the boundary to weaklyenforce Dirichlet constraints. An alternative for imposing constraints is the consistent penalty formulation called the Nitschemethod [32], in which numerical flux terms defined from the primary field stand-in for the multipliers. This method hasserved as one of the inspirations for the Discontinuous Galerkin (DG) method, which has a rich mathematical history [2].A direct connection between the stabilized Lagrange multiplier method and the Nitsche method was established by Stenbergfor the Dirichlet constraint problem [35]. Finally, another relevant approach for treating discontinuous features within ele-ments is the extended finite element method (XFEM) of Belytschko et al. [4]. By combining techniques from XFEM and a DGmethod, Gracie et al. [17] achieved improved convergence properties in the presence of discontinuous enrichments by avoid-ing the parasitic effects of blending elements.

A limitation of the classical Nitsche or GLS approaches [32,3] is that the explicit definition of the stabilization parameter isleft unspecified by the theory. Refinement of these methods over the years for embedded interface and DG methods haverevolved around establishing estimates that provide robust performance under a wide range of interface conditions. A tra-ditional approach proposed by Griebel and Schweitzer [18] employs the solution of a global eigenvalue problem. An earlywork by Brezzi et al. [9] demonstrated that bubble functions could be used to approximate the stabilization parameterfor imposing Dirichlet constraints on the boundary of linear triangular meshes. This technique was extended by Mouradet al. [31] to handle the situation where the Dirichlet interface intersects the finite elements; subsequent refinements in-clude a study employing residual-free bubbles [12] in place of the simple polynomial functions as well as the developmentof closed-form algebraic expressions for linear triangular elements [13]. A bubble stabilization approach is also proposed bySanders et al. [33] for imposing tied constraints across enriched interfaces in linear elasticity.

Although simple approximations or global estimates for the stabilization parameter are satisfactory for most situations,robust estimates become crucial in the presence of sharply varying material properties, anisotropic nonconforming meshes,or degenerate intersections of elements by embedded interfaces. While low values of the parameter can lead to loss of coer-civity and unstable numerical results, high values can yield overly stringent enforcement of the constraints and degradedrepresentations of the interface flux analogous to the penalty method. Therefore, attempts have been made to determinerelationships between the definitions for the stabilization parameter and the numerical flux in the Nitsche method. Forthe Poisson equation [19] and linear elasticity [20], Hansbo and coworkers have proposed an area-weighting scheme forthe numerical flux in an unfitted interface method. Alternatively, a stiffness-weighting approach was utilized by Sanderset al. [34] to alleviate stress-locking in the vicinity of embedded interfaces for the Nitsche method applied to linear elasticity.Recently, a robust version of the Nitsche method was proposed by Annavarapu et al. [1] in which generalized inverse esti-mates for linear simplex elements provided a coupled definition for both the weighted-average numerical flux and the sta-bilization parameter for embedded interface problems.

In this paper, we present a novel approach for deriving a Nitsche interface method from an underlying Lagrange multi-plier method by applying a Variational Multiscale (VMS) approach [22]. We focus on the case in which the interface coincideswith element boundaries, although the methodology is quite general and could be extended to embedded interfaces. Startingfrom a Lagrange multiplier formulation for imposing continuity constraints weakly at the interface, a multiscale decompo-sition is applied to the primary field locally at the interface. Recognizing that the discretization process induces instabilitiesnot present in the continuum problem, we incorporate models for the fine-scale features in order to enhance the stability inthe neighborhood of the interface. By modeling these fine-scale features using edge bubble functions, we obtain consistentresidual-driven terms at the interface that stabilize the mixed primal-multiplier formulation. In the case of the one-sidedDirichlet boundary problem, the present approach is backward compatible to the formulation of Barbosa and Hughes [3] ex-cept that the stabilization parameter is explicitly derived from the fine-scale models. By employing discontinuous functionsfor the Lagrange multipliers along segments of the interface, a definition for the numerical flux emerges that, upon substi-tution into the mixed form, yields a DG interface method where the primary field is the only unknown. Because of the inher-ent similarity in the weak formulations between the Nitsche method and the classical interior penalty DG method [1], wewill refer to these methods interchangeably throughout the discussion in this work.

The methodology presented herein shares features with the bubble stabilization approaches in [9,31,12,33]. However,these works focused mainly on the weak imposition of Dirichlet boundary conditions for linear triangle and tetrahedral dis-cretizations. By analyzing the coupled interface problem, we develop a systematic procedure for deriving both the stabiliza-tion parameter as well as a weighted numerical flux that account for discrete geometric or material mismatch locally at theinterface in lieu of the approaches in [19,34,1]. Specifically, we observe excellent numerical performance obtained on non-conforming meshes for the Poisson equation and elasticity with thermal stresses. We wish to emphasize that this procedurecan be applied to general classes of PDEs across the common interface and illustrate the ideas via the coupling of displace-ment-based and mixed form of elasticity. An algorithm is also proposed for automatically defining the fine-scale bubblefunctions for a family of linear and quadratic elements based on the local interface topology. This stabilized interface ap-proach represents a first step toward deriving DG methods for nonlinear PDEs, where it has been observed (e.g. [36]) thatthe stabilization parameter may need to evolve with the nonlinear solution as compared to the linear context in which apriori estimates can be satisfactory [24].

T.J. Truster, A. Masud / Comput. Methods Appl. Mech. Engrg. 268 (2014) 194–224 195


The remainder of the paper is organized as follows. In Section 2, a general framework employing VMS ideas is presentedfor deriving DG methods to couple nonconforming meshes as well as different PDEs across discrete interfaces, wherein con-sistent expressions for the stabilization parameter and weighted numerical flux emerge. Specialization of the framework torepresentative governing equations follows in Section 3. An investigation of various representations of the fine-scale fieldslocalized to the interface is carried out in Section 4. A series of numerical studies for the Poisson equation, linear elasticity,and combined pure-displacement and mixed elasticity are presented in Section 5. Finally, Section 6 contains concluding re-marks concerning the multiscale approach.

2. Stabilized method for coupling different PDEs across interfaces

Consider a domain X � Rnsd containing two open regions Xð1Þ and Xð2Þ such that �X ¼ �Xð1Þ [ �Xð2Þ, and the two subdomainsshare a common interface CI; see Fig. 1. Let Vð1Þ and Vð2Þ be real Hilbert spaces associated with each respective region, andalso let W be a real Hilbert space defined over the interface CI . The dual spaces are denoted by V0ðaÞ and W0 along with the

duality pairings �; �ð ÞxðaÞ and �; �h ic on domains xðaÞ # XðaÞ and c# CI , where the parameter a takes the value 1 or 2 throughout

the following section. In each region XðaÞ, the primary fields uð1Þ;uð2Þ� �

2 Vð1Þ � Vð2Þ are governed by a different linear PDE asfollows:

AðaÞuðaÞ þ f ðaÞ ¼ 0 in XðaÞ ð1ÞuðaÞ ¼ 0 on CðaÞ ð2Þ

where AðaÞ : VðaÞ ! V0ðaÞ are linear (differential) operators, f ðaÞ 2 V0ðaÞ are source terms, and CðaÞ ¼ @XðaÞ n CI is the portion ofthe region boundary @XðaÞ excluding the interface. These equations are to be interpreted in the sense of distributions [3]. Forsimplicity of presentation, we have assumed homogeneous Dirichlet boundary conditions for both regions; other types ofboundary conditions can be easily treated and do not impact the derivations of the interface terms in the following sections.We require that AðaÞ are such that the fields uðaÞ have the same physical connotation (e.g. displacement, velocity, or concen-tration) throughout X and that the local Eq. (1) are well-posed in the sense of the Lax–Milgram theorem; otherwise, theoperators can be quite general. For example, the operators could both correspond to linear elasticity but represent regionswith different material constants; alternatively, separate operators could be used to solve coupled elastoplasticity problemswith an elastoplastic region embedded in an otherwise elastic domain.

To connect the two regions at the interface, the domain interior Eq. (1) are supplemented with the following continuityequations:

Að1Þuð1Þ � k ¼ 0 on CI ð3ÞAð2Þuð2Þ þ k ¼ 0 on CI ð4ÞBð1Þuð1Þ ¼ Bð2Þuð2Þ on CI ð5Þ

The Lagrange multiplier field k 2 W plays the role of the flux across the interface. The linear operators AðaÞ : VðaÞ ! W are theflux operators associated with the interior operators AðaÞ such that the following integration by parts formulas hold:

w;AðaÞv� �

XðaÞ ¼ �aðaÞXðaÞ

w;vð Þ þ bðaÞCI

�AðaÞv;w

�8v ;w 2 VðaÞ ð6Þ

where aðaÞXðaÞ�; �ð Þ are bilinear operators on VðaÞ � VðaÞ. The trace operators BðaÞ : VðaÞ ! W0 map the primary field uðaÞ to its trace

on CI for pairing with the multiplier l:

l;BðaÞvD E

CI

¼ bðaÞCIl;vð Þ 8 l;v

� �2 W � VðaÞ ð7Þ

and bðaÞCI�; �ð Þ are bilinear operators on W �VðaÞ.

Fig. 1. Single domain with imposed interface.

196 T.J. Truster, A. Masud / Comput. Methods Appl. Mech. Engrg. 268 (2014) 194–224


Remark. To illustrate the above definitions, consider the scalar Poisson equation. The functional spaces can be identified as

VðaÞ ¼ H10 XðaÞ� �

¼ v 2 H1 XðaÞ� �

jcov ¼ 0 on CðaÞn o

and W ¼ H�12 CIð Þ, where H1 XðaÞ

� �, H�

12 CIð Þ denote standard Hilbertian-

Sobolev spaces. Additionally, the various differential and trace operators are defined as Að1Þv ð1Þ ¼ r � Að1Þrv ð1Þ� �

; Að2Þv ð2Þ

¼ r � Að2Þrv ð2Þ� �

; Að1Þv ð1Þ ¼ nð1Þ � Að1Þrv ð1Þ; Að2Þv ð2Þ ¼ nð2Þ � Að2Þrvð2Þ; Bð1Þv ð1Þ ¼ cov ð1Þ, and Bð2Þv ð2Þ ¼ cov ð2Þ, wherein

Að1Þ; Að2Þ are symmetric positive definite constitutive tensors, nð1Þ; nð2Þ are the unit outward normals to the respective

boundaries @Xð1Þ; @Xð2Þ as shown in Fig. 1, and co : H1 XðaÞ� �

! H12 @XðaÞ� �

is the trace operator that extends the primary

field from the interior onto the boundary. Finally, the bilinear operators are defined as aðaÞXðaÞ

v ;wð Þ ¼RXðaÞ rv � AðaÞrw

� �dX

and bðaÞCIl;vð Þ ¼

RCI

lcov dC:

With these definitions, the weak form associated with (1)–(5) becomes: Find uð1Þ;uð2Þ; k� �

2 Vð1Þ � Vð2Þ �W such that forall wð1Þ;wð2Þ;l� �

2 Vð1Þ � Vð2Þ �W:

að1ÞXð1Þ

wð1Þ;uð1Þ� �

þ að2ÞXð2Þ

wð2Þ;uð2Þ� �

þ bð1ÞCI�k;wð1Þ� �

þ bð2ÞCIk;wð2Þ� �

¼ wð1Þ; f ð1Þ� �

Xð1Þþ wð2Þ; f ð2Þ� �

Xð2Þð8Þ

bð1ÞCI�l;uð1Þ� �

þ bð2ÞCIl;uð2Þ� �

¼ 0 ð9Þ

In the sequel, we will consider finite element discretizations of this weak form. Because the continuity condition (5) isweakly enforced, the finite element mesh is permitted to be nonconforming along the interface CI . For each a 2 1;2f g, let

T ðaÞ be a partition of region XðaÞ into open, non-overlapping regions XðaÞe

n onðaÞumel

e¼1with associated boundaries CðaÞe

n onðaÞumel

e¼1, where

nðaÞumel is the number of elements. The partitions satisfy the following closure and intersection properties:

[nðaÞumel

e¼1

�XðaÞe ¼ �XðaÞ;\nðaÞumel

e¼1

XðaÞe ¼£ ð10Þ

We denote the union of element interiors and element boundaries by ~XðaÞ and ~CðaÞ, respectively. As such, �XðaÞ ¼ ~XðaÞ [ ~CðaÞ.Finally, we denote the intersection of an element boundary with a portion of the region boundary by adding a subscript, e.g.

CðaÞI;e � CðaÞe \ CI , and the union of all such element boundaries is denoted by CðaÞI . The finite element subspaces VhðaÞ are taken

to be the spaces of complete piecewise continuous polynomials of degree kðaÞ:

Vhð1Þ ¼ uhð1Þ uhð1Þ 2 C0 Xð1Þ� �

\ Vð1Þ; uhð1Þ��Xð1Þe2 Pkð1Þ Xð1Þe

� �for e ¼ 1; . . . ;nð1Þumel

��n oð11Þ

Vhð2Þ ¼ uhð2Þ uhð2Þ 2 C0 Xð2Þ� �

\ Vð2Þ; uhð2Þ��Xð2Þe2 Pkð2Þ Xð2Þe

� �for e ¼ 1; . . . ;nð2Þumel

��n oð12Þ

where in general the polynomial degrees kð1Þ and kð2Þ as well as the master element shape may differ between the two re-gions. At this point, we leave the discrete multiplier spaceWh � W unspecified. With these definitions, the Galerkin form of

(1)–(5) is stated as: Find uhð1Þ;uhð2Þ; khn o

2 Vhð1Þ � Vhð2Þ �Wh such that for all whð1Þ;whð2Þ;lh� �

2 Vhð1Þ � Vhð2Þ �Wh:

að1Þ~Xð1Þwhð1Þ;uhð1Þ� �

þ að2Þ~Xð2Þwhð2Þ;uhð2Þ� �

þ bð1Þ~Cð1ÞI

�k;whð1Þ� �þ bð2Þ~Cð2Þ

I

k;whð2Þ� �¼ whð1Þ; f ð1Þ� �

~Xð1Þþ whð2Þ; f ð2Þ� �

~Xð2Þð13Þ

bð1Þ~Cð1ÞI

�l;uhð1Þ� �þ bð2Þ~Cð2Þ

I

l;uhð2Þ� �¼ 0 ð14Þ

While the continuum formulation (8) and (9) is well-posed, its discrete counterpart (13) and (14) does not inherit the samestability properties, which leads to technical issues when the classical continuous Galerkin approach is applied to the indi-vidual subproblems in (1) and (2) for a 2 1;2f g. Namely, the formulation can be characterized as a mixed field problem, andtherefore the selection of piecewise continuous or discontinuous multipliers as well as the polynomial order has significantimplications on the stability of the discrete problem. In particular, employing combinations which do not satisfy the Babuš-ka–Brezzi condition [7] may lead to unstable or oscillatory numerical results. Namely, the additional multiplier field kh rep-resents additional unknowns that must be solved for, and the stiffness matrix resulting from the mixed system of equation(13) and (14) is indefinite, which is not as easily accommodated by solvers in existing finite element codes.

To avoid these drawbacks, an alternative technique first proposed by Nitsche [32] relies on numerical fluxes to enforcethe continuity condition (5). Thus, no additional fields are introduced, and the resulting stiffness matrix is symmetric posi-tive definite. However, a penalty term is often necessary for stability, and the value of the coefficient is left unspecified by thetheory. Additionally, the representation of the numerical flux using the simple average of gradients has been found to pro-duce inaccurate or unstable results in the presence of highly disparate grid sizes or material properties [19,34,1]. In the fol-lowing sections, we present a consistent derivation of a generalized Nitsche method in which the definition of the penalty



parameter arises naturally. While the relationship between the Nitsche method and stabilized Lagrange multiplier methodshas been established [35], the unique contribution of this work is the presentation of a systematic approach for derivingmethods associated with a wider class of linear PDEs, including the capability of treating different governing equations ineach of the subregions.

2.1. Variational Multiscale approach

The critical link between the Nitsche method and the Lagrange multiplier method is exposed through the lens of the Var-iational Multiscale (VMS) method. The guiding philosophy of this method [22] is that the exact solution field is decomposedinto a coarse-scale part that is representable by the discrete solution space and a fine-scale part that is incapable of beingresolved by the given finite element mesh. Often, the lack of resolution of the fine-scale features by a given discretizationis the major factor in the loss of stability for many numerical techniques. By introducing models to account for such features,stabilized formulations for a variety of other problems have been developed using the VMS approach, for which we cite[16,26–28,11].

In the present context of the Lagrange multiplier formulation given by (8) and (9), we apply a multiscale overlappingdecomposition of the primary field in each region into coarse scales and fine scales:

uð1Þ ¼ uð1Þ þ ~uð1Þ; wð1Þ ¼ wð1Þ þ ~wð1Þ ð15Þuð2Þ ¼ uð2Þ þ ~uð2Þ; wð2Þ ¼ wð2Þ þ ~wð2Þ ð16Þ

We remark that for mixed field problems, a decomposition may not be required for each primary field in order to achieve astable method; see for example the discussions on the Navier–Stokes equations [26], mixed form of elasticity [30], advec-tion–diffusion [27], and turbulence modeling in [11]. No continuity requirements regarding the interface are placed onthe coarse or fine scales. The coarse scales uð1Þ; uð2Þ are associated with finite element spaces Vhð1Þ and Vhð2Þ, respectively,and the fine scales are taken to lie in the complement spaces Vð1Þ n Vð1Þ and Vð2Þ n Vð2Þ. Substituting (15) and (16) into the gen-eral weak form (8) and (9) and using the linearity of the bilinear forms, we obtain the multiscale weak form for the interfaceproblem:

Coarse-scale problem

að1Þ~Xð1Þwð1Þ; uð1Þ þ ~uð1Þ� �

þ að2Þ~Xð2Þwð2Þ; uð2Þ þ ~uð2Þ� �

þ bð1Þ~Cð1ÞI

�k; wð1Þ� �þ bð2Þ~Cð2Þ

I

k; wð2Þ� �

¼ wð1Þ; f ð1Þ� �

~Xð1Þþ wð2Þ; f ð2Þ� �

~Xð2Þð17Þ

bð1Þ~Cð1ÞI

�l; uð1Þ þ ~uð1Þ� �

þ bð2Þ~Cð2ÞI

l; uð2Þ þ ~uð2Þ� �

¼ 0 ð18Þ

Fine-scale problem

að1Þ~Xð1Þ~wð1Þ; uð1Þ þ ~uð1Þ� �

þ að2Þ~Xð2Þ~wð2Þ; uð2Þ þ ~uð2Þ� �

þ bð1Þ~Cð1ÞI

�k; ~wð1Þ� �þ bð2Þ~Cð2Þ

I

k; ~wð2Þ� �

¼ ~wð1Þ; f ð1Þ� �

~Xð1Þþ ~wð2Þ; f ð2Þ� �

~Xð2Þð19Þ

At this point, our objective is to derive an analytical expression for the fine scales in terms of the Euler–Lagrange equationsfor the coarse-scale fields uðaÞ and k. Crucially, the proposed method relies upon the assumptions that the fine-scale field islocalized to the elements adjacent to CI and does not vanish along CI . Upon substitution of this expression into the coarse-scale problem, the fine-scale effects at the interface produce a stabilized mixed formulation. With the additional assumptionof discontinuous Lagrange multipliers, a stabilized weak form for the primary fields uðaÞ alone is obtained in which consistentdefinitions emerge for the weighted numerical flux and the penalty parameter.

Remark. Traditionally, the fine scales are assumed to vanish on element boundaries in order to simplify the theoreticalderivations as well as the subsequent implementation (see e.g. [28]). However, fine-scale contributions on elementboundaries have been studied in other contexts, such as in the design of stabilized methods for the reaction–diffusionproblem [16] and in a posteriori error estimation techniques for stabilized methods [28].

2.2. Modeling of fine scales at the interface

To facilitate the derivation of the fine-scale models, we first introduce notation associated with the partition of the inter-face CI into segments that are induced by the coarse-scale discretizations T ð1Þ and T ð2Þ. A segment cs is defined as the non-

empty intersection between the boundaries of a single pair of elements Xð1Þes;Xð2Þes

� �2 T ð1Þ � T ð2Þ from the adjoining regions:

cs ¼ int Cð1Þes\ Cð2Þes

� �ð20Þ

An example interface segment is shown in Fig. 2 for a quadrilateral element mesh; the meaning of the red/dashed subregions

xð1Þs and xð2Þs will be described below. We consider the simple case where segments are lines for two-dimensional discret-izations and general polygons for three-dimensional discretizations. The set of all such segments T I ¼ csf g

nsegs¼1, where nseg is



the number of segments, serves as a covering for interface CI:Snseg

s¼1�cs ¼ �CI . The union of all segments is denoted by ~CI . Finally,

let N ð1Þ ¼ Xð1Þes

n onseg

s¼1and N ð2Þ ¼ Xð2Þes

n onseg

s¼1denote the sets of elements from each region that are adjacent to ~CI .

With these definitions in hand, we begin by assuming that the fine scales are non-zero only within elementsN ðaÞ adjacentto the interface and along element boundaries that intersect the interface CI:

~uðaÞ ¼ ~wðaÞ ¼ 0 on ~CðaÞ n ~CI; ~uðaÞ ¼ ~wðaÞ ¼ 0 in XðaÞe 2 TðaÞ n N ðaÞ ð21Þ

Next, we approximate the fine-scale fields using edge bubble functions. However, the support of the bubble bðaÞs is taken to bethe tributary area or sector xðaÞs # XðaÞes

which respects the interface segment cs, namely @xð1Þs \ ~CI ¼ @xð2Þs \ ~CI ¼ cs; a similarconstruction was also adopted in [9,31]. An example of an interface segment and associated sectors is depicted in Fig. 2,where the sectors are indicated by the red/dashed lines within the respective element. The characteristics of the bubble func-tions and sectors are described in detail in Section 4 for the various element types; an example is shown in Fig. 3. Presently,we remark that the weak continuity constraint (18) implies that the bubble functions need not conform along cs:bð1Þs

��cs

– bð2Þs

��cs

. In summary, the fine scales in the neighborhood of segment cs are represented as follows:

~uðaÞ��xðaÞs¼X~nðaÞk¼1

bðaÞsk bðaÞs xð ÞeðaÞk ; ~wðaÞ��xðaÞs¼X~nðaÞl¼1

gðaÞsl bðaÞs xð ÞeðaÞl ð22Þ

where summation is not implied on a, ~nðaÞ is the number of components of the vector-valued fine-scale field ~uðaÞ, and

eðaÞk

n o~nðaÞ

k¼1is a set of linearly-independent unit vectors spanning R

~nðaÞ . Adopting this representation enables the separation

of (19) into a series of local problems associated with segments cs 2 T I:

að1Þxð1Þs

~wð1Þ; ~uð1Þ� �

þ að2Þxð2Þs

~wð2Þ; ~uð2Þ� �

¼ bð1Þcsk; ~wð1Þ� �

þ bð2Þcs�k; ~wð2Þ� �

� að1Þxð1Þs

~wð1Þ; uð1Þ� �

� að2Þxð2Þs

~wð2Þ; uð2Þ� �

þ ~wð1Þ; f ð1Þ� �

xð1Þs

þ ~wð2Þ; f ð2Þ� �

xð2Þs

for s ¼ 1; . . . ;nseg ð23Þ

Fig. 2. Interface segment cs .

Fig. 3. Sample edge bubble function.



Remark. The representation for the fine scales given by (22) is fairly simple because the same bubble function is assumed tocharacterize each component. More elaborate models could be proposed with different functions for each component, suchas approximations using residual-free bubbles in the case of vector field problems.

Requiring that the resulting equations be satisfied independently for all values of gðaÞsl , we can solve for the value of bðaÞsk

independently in sector xðaÞs in terms of the coarse-scale residual and the Lagrange multipliers:

bðaÞsk ¼X~nðaÞl¼1

aðaÞxðaÞs

bðaÞs eðaÞl ; bðaÞs eðaÞk

� ��1bðaÞs eðaÞl ;AðaÞuðaÞ þ f ðaÞ� �

xðaÞs

þbðaÞcs

�1ð Þa�1k� AðaÞuðaÞ; bðaÞs eðaÞl

� �ið24Þ

where the term aðaÞxðaÞs


� ��1is the inverse of the matrix of coefficients generated from varying the subscripts k and

l from 1 to ~nðaÞ, and the term �1ð Þa�1 provides the proper sign for the multiplier on each side of the interface. Additionally,

integration by parts has been applied to the terms aðaÞxðaÞs

~wðaÞ; uðaÞ� �

accounting for assumption (21) within formula (6). This

leads to the residual of the Euler–Lagrange equations (1) restricted to the element interior rather than the integrated by partsform of the residual.

We now introduce three assumptions to further simplify (24). First, the residual of the coarse-scales in the element inte-

rior is assumed to be nearly orthogonal to the fine-scale bubble functions bðaÞs , which is equivalent to representing the fine

scales using residual-free bubbles [6,8,12]. While we do not strictly enforce the condition bðaÞs ¼ bRFB, we argue that the

assumption bðaÞs eðaÞl ;AðaÞuðaÞ þ f ðaÞ� �

xðaÞs

� 0 is valid because the residual is expected to be small for stable numerical methods

and also to decrease as the mesh is refined. Second, we focus on the trace of the fine scales along the interface and make thefollowing approximation, which is analogous to employing the average value of the bubble on segment cs:

BðaÞ~uðaÞ��cs

¼ BðaÞX~nðaÞk¼1

bðaÞsk bðaÞs xð ÞeðaÞk

h i !¼X~nðaÞk¼1

bðaÞsk BðaÞ bðaÞs xð ÞeðaÞk

h i�X~nðaÞk¼1

bðaÞsk

XnWm¼1

~BðaÞkmem

" #ð25Þ

~B að Þkm ¼ meas csð Þ½ ��1 em;B

að Þ b að Þs e að Þ

k

� �D Ecs

ð26Þ

where emf gnWm¼1 is a set of linearly-independent spatially-uniform unit vectors inW0 and nW is the number of components of

the vectors in W0. For example, in the case of linear elasticity discussed in Section 3.2 we have ~nðaÞ ¼ nW ¼ nsd because thevectors in the space of fine scales and multipliers have the same dimension. For problems in which only normal continuity ofvectors is imposed, such as Darcy flow [23], these dimensions may be distinct. Expression (25) is obtained by employing therepresentation (22) and the linearity of the trace operators BðaÞ. Third, the boundary residual �1ð Þa�1

k� AðaÞuðaÞ 2 W is takenoutside of the bilinear form bðaÞcs

�; �ð Þ by applying a projection operator PW0 :W !W0, which is equivalent to applying themean value theorem [26,28]. Combining these steps, the analytical expression for the trace of the fine scales on the interfaceis given by:


¼ sðaÞs PW0 �1ð Þa�1k� AðaÞuðaÞ

h ið27Þ

where the stabilization tensor sðaÞs is expressed as follows:

sðaÞs ¼X~nðaÞk;l¼1

XnWm;n¼1

meas csð Þ½ �~BðaÞkm~skl

~BðaÞln em enð Þ ð28Þ

~skl½ � ¼ aðaÞxðaÞs


� � �1

ð29Þ

Following the approach adopted in related applications of the Variational Multiscale approach in deriving stabilized mixedmethods [26,28], we implicitly treat the projection operator PW0 as the identity and drop its explicit appearance in the sub-sequent derivations.

By employing clearly delineated modeling assumptions, we have arrived at models for the fine scales on each side of the

interface given by (27) that are driven by the boundary residual of the coarse scales. Crucially, the stability tensors sðaÞs

implicitly account for many of the distinguishing continuum and discrete characteristics at the interface through the inte-

gration of the fine-scale bubbles bðaÞs and the bilinear forms aðaÞxðaÞs�; �ð Þ, namely: (i) element size, aspect ratio, shape, and poly-

nomial order; (ii) material properties; and (iii) partial differential operators. These stabilization tensors will have asignificant impact on the form of the numerical flux and penalty parameters derived in the following section. In particular,features naturally emerge that overlap with the area weighting and material weighting techniques proposed in the literaturethat have been shown to yield robust numerical performance [19,34,1].



Remark. The derivations in this section share common features with the bubble stabilization approaches [9,31,12,33]developed for the imposition of Dirichlet constraints on embedded interfaces in the context of elliptic problems. However,the key distinctions between these approaches and the present developments are that a mean value projection is applied toextract the boundary residual from the integral expression in (27) and that the average value of the bubble function isemployed in the final fine-scale model.

Remark. For general vector-valued PDEs, the stabilization tensor obtained from (28) will not be diagonal but instead willinduce coupling between the components of the primary field accounting for the local interface geometry and material prop-erties. This coupling does not introduce complications into the derivations that follow. However, for computational economythe full tensor sðaÞs can be approximated numerically as a diagonal matrix by dropping the coupling terms [26,11].

2.3. Stabilized interface formulation

To obtain the stabilized mixed weak form, we now embed the fine-scale model (27) into the coarse-scale problem (17)and (18). Presently, we require the definition of the adjoint operators AðaÞ : VðaÞ ! V0ðaÞ and AðaÞ : VðaÞ ! W to accommodatethe expression for ~uðaÞ in (17):

aðaÞXðaÞ

w;vð Þ ¼ � AðaÞw;v� �

XðaÞ þ bðaÞCIv ;AðaÞw� �

8v;w 2 VðaÞ ð30Þ

To agree with the assumptions in the preceding section whereby the significant fine-scale contribution derived in (27) islocalized to the boundary, we neglect the element interior integrals AðaÞw;v

� �~Xa

and retain only the boundary integral.

Carrying out the substitution, we arrive at the stabilized mixed weak form for the system of abstract linear PDEs:

að1Þ~Xð1Þwð1Þ; uð1Þ� �

þ að2Þ~Xð2Þwð2Þ; uð2Þ� �

� swt; kh i~CIþ Að1Þwð1Þ; sð1Þs k� Að1Þuð1Þ

h iD E~CI

þ Að2Þwð2Þ; sð2Þs �k� Að2Þuð2Þh iD E

~CI

¼ wð1Þ; f ð1Þ� �


~Xð2Þð31Þ

� l; sut� �

~CI� l; sð1Þs k� Að1Þuð1Þ

h iD E~CI

þ l; sð2Þs �k� Að2Þuð2Þh iD E

~CI

¼ 0 ð32Þ

where we have introduced the jump operator s � t mapping Vð1Þ � Vð2Þ ! W0, which in the present setting is defined assut ¼ Bð1Þuð1Þ � Bð2Þuð2Þ. Because the Lagrange multiplier field is associated with domain Xð1Þ according to (3), namelyk ¼ Að1Þuð1Þ, the sign of the jump term sut depends upon the ordering of domains Xð1Þ and Xð2Þ. However, the final stabilizedform is shown to be independent of this ordering.

Through the incorporation of the stabilizing terms obtained via the fine-scale models, arbitrary combinations for the pri-mary and multiplier fields are admissible in the weak form (31) and (32). To complete the derivation of a stabilized primalinterface formulation analogous to the DG method, we focus on the continuity equation (32) and introduce the assumption

thatW ¼ L2 ~CI

� �h inW, namely the space of discontinuous L2 functions on CI . This approximation allows us to obtain a closed-

form expression for the Lagrange multiplier field k on each segment:

kjcs¼ dð1Þs PW Að1Þuð1Þ

� �� dð2Þs PW Að2Þuð2Þ

� �h i��cs

� ss PWsutð Þjcsð33Þ

where the tensorial quantities are derived from (32) and termed as follows:

ss ¼ sð1Þs þ sð2Þs

� ��1; dðaÞs ¼ sss

ðaÞs ð34Þ

Also, the projection operator PW , which in general mapsW0 intoW, reduces to the identity sinceW ¼W0 ¼ L2 ~CI

� �h inW(i.e.,

PW is surjective as well as injective). Observe that by definition dð1Þs þ dð2Þs ¼ I, where I is the second-order identity tensor.From (33), we extract the definition of the numerical flux over segment cs as the weighted average of the flux from theadjoining regions:

Auf g ¼ dð1Þs Að1Þuð1Þ� �

� dð2Þs Að2Þuð2Þ� �

ð35Þ

The minus sign in (35) reflects the opposite signs of the primal flux from each region XðaÞ according to (3) and (4). For thespecial case that dð1Þs ¼ dð2Þs ¼ 1

2 I, equation reverts to the simple average of the fluxes from both domains that is commonlyemployed in DG methods. However, weighted averages have been found to be more robust for disparate materials or ele-ment sizes [19,34,1]. Herein, the weighting is consistently determined through the fine-scale approximation, and its perfor-mance will be assessed from the numerical studies presented in Section 5. A similar expression to (33) also holds for theadjoint operators.

Finally, substituting (33) and (34) into (31) and regrouping terms, we obtain the stabilized interface formulation for a sys-tem of PDEs in the primary field alone:





� swt; Auf gh i~CI� Awf g; suth i~CI

þ swt; sssuth i~CI� sAwt; dssAuth i~CI

¼ wð1Þ; f ð1Þ� �


~Xð2Þð36Þ

The first three interface integrals in (36) are form-equivalent to the classical DG or symmetric interior penalty method up tothe numerical flux given by (35) and the penalty parameter given by (34). The fourth interface integral involves the product

of flux jump terms defined as sAut ¼ Að1Þuð1Þ þ Að2Þuð2Þ and an inverse penalty parameter obtained through straightforward

algebraic manipulation as ds ¼ sð1Þs dð2Þs ¼ sð2Þs dð1Þs ¼ sð1Þs

� ��1þ sð2Þs

� ��1 �1

, which holds for arbitrary tensors sð1Þs and sð2Þs .

While this term is not commonly used in single field DG methods, it does appear in DG methods for mixed field problemssuch as Darcy flow [23]. These four interface terms model the effects of the Lagrange multipliers and fine-scale fields whichno longer explicitly appear in (36).

A number of properties and remarks concerning the above formulation are summarized in the following sections.

2.3.1. Proof of variational consistencyOur first observation is that the formulation given by (36) is consistent with the governing Eqs. (1)–(5). Let

uð1Þ;uð2Þ� �

2 Vð1Þ � Vð2Þ be such that (36) holds for all wð1Þ;wð2Þ� �

2 Vð1Þ � Vð2Þ. Applying the integration by parts formula (6)and using the jump and flux definitions (35) and (37), the weak form (36) is equivalent to the following:X

a¼1;2

� wðaÞ;AðaÞuðaÞ þ f ðaÞ� �

XðaÞþ bðaÞCI

AðaÞuðaÞ;wðaÞ� �h i

� Bð1Þwð1Þ; I � dð2Þs

� �Að1Þuð1Þ� �D E

~CI

þ Bð1Þwð1Þ; dð2Þs Að2Þuð2Þ� �D E

~CI

þ Bð2Þwð2Þ; dð1Þs Að1Þuð1Þ� �D E

~CI

� Bð2Þwð2Þ; I � dð1Þs

� �Að2Þuð2Þ� �D E

~CI

þ sTs Bð1Þwð1Þ � dð1ÞTs Að1Þwð1Þ; sut

D E~CI

� sTs Bð2Þwð2Þ � dð2ÞTs Að2Þwð2Þ; sut

D E~CI

� dTs Að1Þwð1Þ; sAut

D E~CI

� dTs Að2Þwð2Þ; sAut

D E~CI

¼ 0 ðT1Þ

where we have employed (34) to obtain dð1Þs ¼ I � dð2Þs and dð2Þs ¼ I � dð1Þs , and the transpose of a tensor is denoted by �ð ÞT . Usingthe definition of the bilinear forms bðaÞCI

�; �ð Þ from (7), we regroup the second terms of the summation with the first four inter-face integrals to obtain:X

a¼1;2

� wðaÞ;AðaÞuðaÞ þ f ðaÞ� �

XðaÞ

h iþ sT

s Bð1Þwð1Þ � dð1ÞTs Að1Þwð1Þ; sutD E

~CI

� sTs Bð2Þwð2Þ � dð2ÞTs Að2Þwð2Þ; sut

D E~CI

þ dð2ÞTs Bð1Þwð1Þ � dTs Að1Þwð1Þ; sAut

D E~CI

þ dð1ÞTs Bð2Þwð2Þ � dTs Að2Þwð2Þ; sAut

D E~CI

¼ 0 ðT2Þ

Finally, invoking the arbitrariness of the weighting functions wð1Þ;wð2Þ yields Euler–Lagrange equations that correspond toEqs. (1)–(5):

AðaÞuðaÞ þ f ðaÞ ¼ 0 in XðaÞ ðT3ÞAð1Þuð1Þ þ Að2Þuð2Þ ¼ 0 on CI ðT4ÞBð1Þuð1Þ � Bð2Þuð2Þ ¼ 0 on CI ðT5Þ

This implies that uð1Þ; uð2Þ is a solution of (1)–(5) in the sense of distributions, which completes the proof. We remark that byreversing the integration by parts using (30), the adjoint consistency of (36) can be proven through direct analogy with thepreceding steps. h

2.3.2. Salient features of the stabilized interface formulationThe following observations summarize the key features of the interface formulation (36):

(1) The stabilization parameters ss and ds and the weighting coefficients dðaÞs for the numerical flux are analytical functionsof the fine-scale stabilization tensors sðaÞs that are in turn tightly coupled to the local interface characteristics as wasremarked in Section 2.2. Therefore, these parameters naturally encompass aspects of the area weighting and materialweighting techniques proposed in [19,34,21,1] . However, the ability to consistently account for different governingPDEs through these parameters is a unique feature of the present approach.

(2) The flux weights simplify to dð1Þs ¼ dð2Þs ¼ 12 I only in the special case that sector xð1Þs is an isometric affine (rigid-body)

transformation Q of sector xð2Þs ; bð1Þs xð Þ ¼ bð2Þs Q xð Þð Þ for all x 2 xð1Þs (usually implying the polynomial degrees

kð1Þ ¼ kð2ÞÞ, and the differential operators (and associated material properties) are identical such that

að1Þxð1Þs

bð1Þs eð1Þl ; bð1Þs eð1Þk

� �¼ að2Þ

xð2Þsbð2Þs eð2Þl ; bð2Þs eð2Þk

� �. These are precisely the assumptions of homogeneity (shape-regularity)

that are invoked when postulating the classical definition of the numerical flux in DG methods [2].



(3) This derivation clearly exposes the link between the classical Discontinuous Galerkin method and an underlyingLagrange multiplier method for weak imposition of continuity constraints. This mathematical connection gives furthercredibility to the rich history of positive performance exhibited by such formulations [2].

(4) The modeling assumptions applied to the fine scales admit nonconforming meshes and higher-order polynomial func-tions for the coarse scales, as demonstrated by the numerical results in Section 5.

(5) The multiscale derivation provides a systematic procedure for developing Discontinuous Galerkin methods for a wideclass of linear PDE and also serves as an outline for extending such methods to nonlinear problems.

Remark. There is similarity between the form of the stability parameter in (36) and the bubble stabilized methods forenriched interfaces present in the literature [9,31,12,33]. However, in those studies the penalty term involving the primaljump is represented as a product of integrals rather than an integral of a product as is contained in (36). This noteworthydifference arises due to the use of the average bubble function in expression (27). The form of the penalty term presentin (36) is more appropriate for extension to nonlinear problems where the interface conditions can vary from point to point,such as contact with friction in the case of solid mechanics applications.

Remark. As mentioned previously, the sign of the jump term depends upon the ordering of regions Xð1Þ and Xð2Þ. However,this is also true for the definition of the numerical flux (35) due to the sign of k that is tied to the primal fluxes because of (3)and (4). Thus, the product of the numerical flux and the jump term is invariant under a renumbering of the regions. Otherdefinitions for the jump and flux can be defined which are invariant (see e.g. [7,29]). However, the integration by parts for-mulas for these terms are less transparent, and a sign convention must still be adopted for the Lagrange multipliers.

Remark. Similar to the viewpoint taken in [28], the fine-scale fields in the present derivations can be viewed as a measure ofthe numerical error, arising from the inability of the shape functions on the nonconforming mesh to resolve all the featuresof the true solution. Indeed, under the present approximation of the fine scales and the multipliers, combining (27) and (33),and the definition of sut leads to the identity s~ut � �sut, which gives credence to the interpretation of the fine scales as thediscretization error. Thus, the fine scales are expected to be on the same order as the primal jump at the interface, which wasconfirmed by our numerical studies. The reader is referred to [28] for further understanding of the role of fine scales.

3. Specialization to particular governing equations

To make the preceding developments more concrete, we now demonstrate the specialization of the stabilized interfaceformulation to various model problems, first considering a common PDE for both subdomains and then discussing the multi-PDE case.

3.1. Poisson equation

The first example is the weighted Poisson equation, for which the corresponding differential operators and bilinear formsfor the concentration field uðaÞ in each region XðaÞ are defined as follows:

aðaÞXðaÞ

wðaÞ;uðaÞ� �

¼Z

XðaÞrwðaÞ � AðaÞruðaÞ

� �dX ð37Þ

AðaÞuðaÞ ¼ AðaÞuðaÞ ¼ nðaÞ � coAðaÞruðaÞ ð38ÞBðaÞuðaÞ ¼ couðaÞ ð39Þ

The fine-scale models at the interface are obtained by substituting these definitions into Eq. (27):

~uðaÞ��cs¼ sðaÞs �1ð Þa�1k� nðaÞ � AðaÞruðaÞ

� �h ið40Þ

where the stabilization parameter sðaÞs is given by the following simplified expression for this scalar field problem:

sðaÞs ¼ meas csð Þ½ ��1Z

cs

bðaÞs dC

!2

aðaÞxðaÞs

bðaÞs ; bðaÞs

� ��1ð41Þ

Embedding the expression for the fine scales into the corresponding coarse-scale problem and carrying out the steps in Sec-tion 2.3 leads to a stabilized interface formulation for the Poisson problem:



� swt; n � Aruð Þf gh i~CI� n � Arwð Þf g; suth i~CI

þ swt; sssuth i~CI

¼ wð1Þ; f ð1Þ� �

~Xð1Þ þ wð2Þ; f ð2Þ� �

~Xð2Þ ð42Þ

in which the concentration jump is defined as sut ¼ uð1Þ � uð2Þ and the numerical flux follows directly from (33):



n � Aruð Þf g ¼ n � dð1Þs Að1Þruð1Þ þ dð2Þs Að2Þruð2Þ� �

ð43Þ

where the unit normal vector n ¼ nð1Þ ¼ �nð2Þ. Also, the stability parameter and weighting coefficients follow from (34) as

ss ¼ sð1Þs þ sð2Þs

� ��1and dðaÞs ¼ sssðaÞs . For simplicity, the primal flux jump term has been neglected.

Remark. As this problem has been cast in the general framework of Section 2.3, the result regarding variational consistencypresented in Section 2.3.1 extends to the present case; this proof is analogous to the result for the Poisson and bi-harmonicequations contained in [14]. The formulation (43) is identical to the method presented in [13] up to the definition of thestabilization terms and flux weighting, which have been generalized to encompass other element types beyond themathematical analysis presented therein.

3.2. Linear elasticity

As a second example, we consider the problem of coupling linear elastic domains with nonmatching meshes and possiblydissimilar material properties, for which the bilinear form and boundary operators in (6) and (7) take the following form:

aðaÞXðaÞ


¼Z

XðaÞe wðaÞ� �

: CðaÞe uðaÞ� �h i

dX ð44Þ

AðaÞuðaÞ ¼ AðaÞuðaÞ ¼ nðaÞ � co CðaÞe uðaÞ� �h i

ð45Þ

BðaÞuðaÞ ¼ couðaÞ ð46Þ

where uðaÞ 2 H1 XðaÞ� �� nsd

is the displacement field restricted to region XðaÞ; e �ð Þ ¼ 12 r �ð Þ þ r �ð Þð ÞTh i

is the symmetric gradi-ent operator, and CðaÞ is a fourth-order symmetric positive definite tensor of material moduli. From (44), we identify thestress tensor as rðaÞ ¼ CðaÞe uðaÞ

� �and the strain tensor as eðaÞ ¼ e uðaÞ

� �. Substituting these definitions into the general Eq.

(27) leads to the following expression for the fine scales:


¼ sðaÞs �1ð Þa�1k� nðaÞ � rðaÞ uðaÞ

� �h ið47Þ

where k 2 L2 CIð Þ� �nsd

has the physical connotation of the interface traction derived from domain Xð1Þ, and the expression forsðaÞs can be simplified to give:

sðaÞs ¼Xnsd

k;l¼1

meas csð Þ½ ��1Z

cs

bðaÞs dC

!2

aðaÞxðaÞs


� � �1

ek elð Þ ð48Þ

Proceeding along the lines of the derivation in Section 2.3, the displacement jump follows simply as sut ¼ uð1Þ � uð2Þ, and wedefine the numerical flux according to (33) as follows:

n � Ce uð Þð Þf g ¼ dð1Þs nð1Þ � Cð1Þe uð1Þ� ��

� dð2Þs nð2Þ � Cð2Þe uð2Þ� ��

ð49Þ

where the weighting tensors dðaÞs are defined as:

dðaÞs ¼ sssðaÞs ; ss ¼ sð1Þs þ sð2Þs

� ��1 ð50Þ

Remark. While the stabilization tensors sðaÞs are usually diagonally-dominant for elements with acceptable aspect ratios,they are not diagonal for triangular or distorted quadrilateral meshes [26]. Thus, the numerical flux �f g involves a generallinear combination of the traction components from the adjoining regions. If desired, the off-diagonal terms of the matrices

sðaÞs

h icould be dropped in order to simplify the calculations, as mentioned in a remark in Section 2.2.

Substituting these results into (36) and neglecting the traction jump term, we obtain the stabilized interface formulationfor linear elasticity:



� swt; n � Ce uð Þð Þf gh i~CI� n � Ce wð Þð Þf g; suth i~CI

þ swt; sssuth i~CI

¼ wð1Þ; f ð1Þ� �


~Xð2Þð51Þ

Up to the definition of the penalty term and the weighted numerical flux, this formulation is form-identical to the DG meth-od proposed in [29].



3.3. Accommodation of residual stresses and strains

Next, our aim is to extend to the interface formulation (51) to account for residual stresses of the following form:

�rðaÞ uðaÞ� �

¼ CðaÞ e uðaÞ� �

� T ðaÞcðaÞh i

þ rðaÞi ¼ rðaÞ uðaÞ� �

þ r0ðaÞ ð52Þ

where TðaÞ is a relative temperature field, cðaÞ is a thermal strain tensor often taken to be cðaÞ ¼ gðaÞI with gðaÞ the coefficient ofthermal expansion and I the second-order identity tensor, and rðaÞi is an initial stress tensor. In order to include the effects ofthis modified stress tensor in (51), we introduce the following affine functional:

�aðaÞXðaÞ


¼ aðaÞXðaÞ


þ lðaÞXðaÞ

wðaÞ� �ð53Þ

lðaÞXðaÞ

wðaÞ� �¼Z

XðaÞe wðaÞ� �

: r0ðaÞ dX ð54Þ

The derivations in Sections 2.1–2.3 can be extended in a straightforward manner to admit affine functionals; we suppressthese details. The equations for ~uðaÞ and sðaÞs in Section 3.2 remain valid except that the stress tensor rðaÞ in the expressionfor the fine scales (47) is replaced by �rðaÞ from (52). A modification is also required in the numerical flux term to accountfor the boundary term resulting from integration by parts of the thermal term:

n � �r uð Þf g ¼ n � r uð Þf g þ dð1Þs nð1Þ � r0ð1Þ� �

� dð2Þs nð2Þ � r0ð2Þ� �

¼ n � Ce uð Þð Þf g þ n � r0f g ð55Þ

Substituting (52) and (55) into (51), we arrive at the stabilized interface formulation for linear elasticity with residualstresses:



� swt; n � Ce uð Þð Þf gh i~CI� n � Ce wð Þð Þf g; suth i~CI

þ swt; sssuth i~CI

¼ wð1Þ; f ð1Þ� �


~Xð2Þ� lð1Þ~Xð1Þ

wð1Þ� �� lð2Þ~Xð2Þ

wð2Þ� �

þ swt; n � r0f gh i~CIð56Þ

Remark. The inclusion of the nonstandard term n � r0f g on the right-hand side could be easily overlooked in theimplementation of the classical DG method for elasticity with thermal strains. However, this term arises consistently duringthe present derivations, and the importance of retaining this term is demonstrated in the numerical example studied inSection 5.3.

3.4. Mixed elasticity

As an example involving a mixed field problem, we consider a displacement–pressure formulation for elasticity that iscapable of modeling incompressible materials. The main enhancement beyond the formulation in Section 3.2 is that thestress tensor becomes a function of the kinematic pressure pðaÞ : XðaÞ ! R. Assuming isotropic behavior, the stress–strainconstitutive equation and associated compatibility condition are stated as:

rðaÞ uðaÞ; pðaÞ� �

¼ pðaÞI þ 2lðaÞe uðaÞ� �

in XðaÞ ð57Þr � uðaÞ ¼ pðaÞ=kðaÞ in XðaÞ ð58Þ

where kðaÞ; lðaÞ are the Lame parameters characterizing the material in each region (not to be confused with the Lagrangemultipliers). Substituting (57) into (44) leads to a three-field formulation in terms of u; p; kf g. In order to eliminate the inter-facial Lagrange multipliers, we again follow the procedure presented in Section 2.1–2.3. Presently, a multiscale decomposi-tion is applied only to the displacement field; this simplifies subsequent derivations and has been demonstrated to besufficient for stabilizing mixed-field problems [26,30,28]. Adopting the fine-scale modeling assumptions from Section 2.2,we arrive at the following expression for ~uðaÞ at the interface:


¼ sðaÞs �1ð Þa�1k� nðaÞ � rðaÞ uðaÞ; pðaÞ

� �h ið59Þ

where rðaÞ uðaÞ; pðaÞ� �

is evaluated using (57) and the stabilization tensors sðaÞs are given by:

sðaÞs ¼ meas csð Þ½ ��1Z

cs

bðaÞs dC

!2 ZxðaÞs

lrbðaÞs � rbðaÞs dX

!I þ

ZxðaÞs

lrbðaÞs rbðaÞs dX

" #�1

ð60Þ

Embedding the representation of the fine scales (59) into the associated coarse-scale problem and proceeding along the linesof Section 2.3 to solve for the multiplier field k, we arrive at the stabilized interface formulation for mixed elasticity:



Xa¼1;2

e wðaÞ� �;2lðaÞe uðaÞ

� �� ~XðaÞ þ r � w

ðaÞ;pðaÞ� �

~XðaÞ

�þXa¼1;2

qðaÞ;r � uðaÞ� �

~XðaÞ � qðaÞ;pðaÞ=k� �

~XðaÞ

�� swt; n � r u; pð Þf gh i~CI

� n � r w; qð Þf g; suth i~CIþ swt; sssuth i~CI

¼Xa¼1;2

wðaÞ; f ðaÞ� �

~XðaÞð61Þ

where the numerical flux terms are computed using (49) with the stress tensor rðaÞ ¼ CðaÞe uðaÞ� �

replaced by definition (57).Although the interface formulation (61) accommodates nonconforming discretizations for the displacement and pressure

fields along CI , the discrete function spaces for these two fields must satisfy the Babuška–Brezzi condition associated withthe domain interior terms for each region XðaÞ in order for (61) to be globally stable. To admit equal-order interpolations ofthe displacement and pressure fields, additional domain-based stabilization terms are incorporated into the weak form thatcan be derived by employing a Variational Multiscale approach to the displacement field on element interiors [28,29].Although the fine-scale models employed for stabilizing the mixed displacement–pressure formulation are assumed to van-ish on element boundaries and therefore do not contribute directly to the interface integrals, overlapping contributions fromthe edge and interior bubble functions would be expected on the interior of the elements adjoining the interface. However,we choose to neglect these coupling effects in our implementation for simplicity, which does not upset the consistency of theformulation. Thus, the final form of the proposed interface formulation for mixed elasticity that admits arbitrary interpola-tion combinations across the interface is as follows:X

a¼1;2

e wðaÞ� �;2lðaÞe uðaÞ

� �� ~XðaÞ þ r � w

ðaÞ;pðaÞ� �

~XðaÞ

�þXa¼1;2

qðaÞ;r � uðaÞ� �

~XðaÞ � qðaÞ;pðaÞ=k� �

~XðaÞ

��Xa¼1;2

rqðaÞ þ 2lðaÞr � e wðaÞ� �; sX � rpðaÞ þ 2lðaÞr � e uðaÞ

� � �� ~XðaÞ � swt; n � r u;pð Þf gh i~CI

� n � r w; qð Þf g; suth i~CIþ swt; sssuth i~CI

¼Xa¼1;2

wðaÞ; f ðaÞ� �

~XðaÞ� rqðaÞ þ 2lðaÞr � e wðaÞ� �

; sX � f ðaÞ� �

~XðaÞð62Þ

The form of the interior stabilization tensor sX is adopted from [28]:

sX ¼ be

ZXe

be dCZ

Xe

lrbe � rbe dX� �

I þZ

Xe

lrbe rbe dX �1

ð63Þ

where the bubble function be is supported on the interior of element Xe. We remark that the stabilized weak form (62) isform-equivalent to the DG method proposed in Section 4.7 of [29] except for the crucial distinction that the penalty param-eter and weighting coefficients have been consistently derived according to (50) and (60).

3.5. Multi-PDE model problem: combining pure-displacement and mixed elasticity

As discussed in the beginning of Section 2, the stabilized interface formulation is capable of coupling different physicalgoverning equations across nonmatching interfaces. Therefore, consider as a motivating example a linear elastic domainX that is partitioned into two regions XðaÞ, the first of which is modeled using the pure-displacement constitutive Eq. (44)and the second of which is modeled using mixed elasticity via the relations (57) and (58). Possible reasons for consideringdifferent constitutive models include the ability to simulate (i) composite materials in which one constituent is incompress-ible as well as (ii) localized incompressible plastic flow within an otherwise elastic domain. The obvious computational econ-omy is that the calculation of the pressure field is avoided in the larger region. Combining the results of Sections 3.2 and 3.4,the composite interface formulation is as follows:

e wð1Þ� �

;Cð1Þe uð1Þ� ��

~Xð1Þþ að2Þ~Xð2Þ

wð2Þ; qð2Þ; uð2Þ;pð2Þ� �

� swt; n � r u;pð Þf gh i~CI� n � r w; qð Þf g; suth i~CI

þ swt; sssuth i~CI

¼ wð1Þ; f ð1Þ� �


~Xð2Þ� rqð2Þ þ 2lð2Þr � e wð2Þ� �

; sX � f ð2Þ� �

~Xð2Þð64Þ

where the bilinear form að2Þ~Xð2Þ�; �ð Þ contains all of the domain integral contributions from the left-hand side of the mixed elas-

ticity formulation (62), and the numerical flux terms are evaluated through the following composite definition:

n � r u;pð Þf g ¼ dð1Þs nð1Þ � Cð1Þe uð1Þ� ��

� dð2Þs nð2Þ � rð2Þ uð2Þ;pð2Þ� ��

ð65Þ

Both the stress tensors rðaÞ and the stabilization tensors sðaÞs required to obtain the values for dðaÞs and ss are evaluated accord-ing to the appropriate expressions in Sections 3.2 and 3.4. Thus, the physics of both governing differential operators is con-sistently embedded in the definition of the numerical interface parameters dðaÞs and ss to provide a robust couplingmechanism at the discrete interface. These definitions remove the ambiguity in postulating a Discontinuous Galerkin meth-



od for a coupled system of PDEs. The performance of this multi-PDE elasticity formulation is assessed through a benchmarkstudy in Section 5.4.

Remark. In the preceding developments, the fine scales are viewed as arising due to the nonconforming discreterepresentation of the interface as well as the numerical instabilities in the classical mixed primal-multiplier method. Thus, ~uis treated as variationally embedded within the coarse scales and governed by the same PDE. However, the fine scales couldalso be viewed as a vehicle for accommodating multiscale physical features at the interface. One example is the frictionalcontact of rough surfaces through asperity interactions, where the micro-features on the interface are orders of magnitudesmaller than the macro-structure. A related approach employing multiscale constitutive models for asperity-interactionwithin the DG method is presented in [37].

4. Design of fine-scale bubble functions

Up to this point, we have left the explicit form of the fine-scale bubble functions unspecified. This functional form has asignificant impact both on the robustness of the interface formulation as well as on the computational economy of thenumerical method. The concept of designing fine-scale approximations to attain particular properties has been explored,for example, in the case of the advection–diffusion equation [10,27]. So-called residual-free bubbles have been found to yieldvery accurate results but at a heavy expense of solving local discrete systems within each element [8,12]. In our previouswork [26,28,29], we have seen that simple polynomial bubble functions provide an adequate approximation of the fine scalesto produce a stable and accurate numerical schemes for linear and nonlinear problems [30,27,11]. Therefore, we propose andinvestigate an automatic procedure for generating the edge bubble function for a general sector xðaÞs associated with an inter-face segment cs.

4.1. Definition of interface sectors and polynomial bubble functions

We first discuss a generic procedure for forming the sectors based upon the local interface topology. For the two-dimen-sional case and assuming a planar interface, the segments consist of straight lines formed by intersecting the edges of twoopposing elements. For triangular elements, the sector is formed by connecting the corner node most distant from the inter-face to the two endpoints of the segment; see Fig. 4(a). For quadrilateral elements, the reference coordinates nA and nB of thesegment endpoints are determined from the physical coordinates by inverting the isoparametric mapping. Then, points withthe same n coordinates but on the opposing edge of the quadrilateral g ¼ 1ð Þ are used to define the boundary of the sector;see Fig. 4(b). Using this procedure, the sectors form a disjoint covering of the elements adjoining the interface:

[nseg

s¼1

�xðaÞs ¼[N a

�XðaÞe ;\nseg

s¼1

xðaÞs ¼£ ð66Þ

where N ðaÞ #N ðaÞ is the set of all elements that have an entire edge intersecting the interface.For the three-dimensional case, the overlapping zone of two element faces on opposite sides of the interface is in general

a planar surface with a polygonal boundary. Herein, we refer to this polygon as the parent segment cs, and we define theactual segments cs through a triangulation of the vertices of the parent segment cs, since quadrature rules are well-estab-lished for triangular domains. For tetrahedral elements, the sector associated with segment cs is a (smaller) tetrahedronformed by the vertices of the triangular segment and the node of the element that is most distant from the interface. Forhexahedral elements, the reference coordinates of the segment vertices Pi ¼ ni;gi;�1ð Þ; i ¼ 1;2;3 are projected to the

Fig. 4. Definition of sectors: (a) triangular element; (b) quadrilateral element.



opposing face as P0i ¼ ni;gi;1ð Þ, and these six vertices are joined to form a sector resembling a wedge-shaped element. Bydefining the sectors in this manner, we again obtain a disjoint covering of the elements adjoining the interface satisfyingproperty (66).

Remark. For the three-dimensional case, the triangulation of a parent segment cs containing more than three edges is notunique. While the various possible triangulations will each result in sectors satisfying (66), the bubble functions generatedfrom certain triangulations may lead to higher quality fine-scale approximations than generated by others. The effect ofvarious triangulation patterns on the robustness of the interface formulation will be thoroughly investigated in future work.

Next, we define template edge bubble functions for each standard two-dimensional element type, which are listed in Ta-ble 1. In the element type abbreviations, the letter designates the shape of the element (‘‘T’’ for triangle and ‘‘Q’’ for quad-rilateral), and the number refers to the number of nodes on the element. The coordinates n;gð Þ are with respect to the localcoordinate system engendered by the associated sector xðaÞs such that the bubble vanishes on all boundaries of the sectorexcept for the interface segment cs, where the n-axis is taken parallel to the segment in all cases. The polynomial functionfor each element type is specified as at least one degree higher than the functions associated with the coarse scales or finiteelement shape functions. This condition ensures that the multiscale function spaces remain linearly independent. Addition-ally, the use of distinct bubbles for linear and quadratic elements implies that unequal values will be obtained for the sta-bilization tensor sðaÞs even when two such elements have the same size and shape, thereby accounting for the distinctcharacter of linear versus quadratic interpolations. By employing simple polynomial representations of ~u, the numericalevaluation of the stabilizing parameters incurs minimal added expense on the computation of the interface contributionsduring the assembly process.

4.2. Comparison of residual-free and polynomial bubble functions

To assess the validity of these modeling assumptions for the fine scales, we consider the Poisson equation and investigatethe value of sðaÞs obtained from polynomial bubbles and residual-free bubbles for linear triangular elements of various aspectratios. The topic of residual-free bubbles has been investigated for a variety of problem classes, including its relationship tothe Variational Multiscale method [8] and its application to embedded interface problems [12]. Presently, the term ‘‘residual-free bubble’’ refers to the exact solution of the localized fine-scale problem (23) specialized to the Poisson equation. For thecase of linear triangular shape functions, piecewise constant Lagrange multipliers, and vanishing source term, this equationreduces to solving the following system on each sector along the interface:

r � AðaÞr~uðaÞ� �

¼ 0 in xðaÞs ð67Þ

~uðaÞ ¼ 0 on @xðaÞs n cs ð68Þ

nðaÞ � AðaÞr~uðaÞ� �

¼ 1 on cs ð69Þ

Since this problem is also infinite dimensional, we instead seek a numerical approximation ~uhðaÞ 2 ~Vh that is computed on asubmesh of cells within the sector xðaÞs through the solution of the following discretized weak form:

aðaÞxðaÞs

~whðaÞ; ~uhðaÞ� �¼ ~whðaÞ;1� �

cs8~wðaÞs 2 ~Vh ð70Þ

Once obtained, the discrete fine-scale field ~uhðaÞ is directly substituted in place of bðaÞs in (41) to evaluate the stability param-eter sðaÞs . In the study that follows, we employ a discretization of 64 uniform triangular cells along each edge of the sector xðaÞs

unless stated otherwise.

Remark. Since we are directly solving the fine-scale problem, the assumption that the interior residual is neglected isdirectly enforced, i.e. the fine scale is enforced to be orthogonal to the coarse scale. Therefore, the residual-free bubble isexpected to give more accurate results for the stabilized interface method. Indeed, studies on the distorted meshes in thenumerical section indicate that this is the case. However, the solution of a discrete problem within each sector of theinterface becomes computationally prohibitive as the number of sectors grows. Therefore, we use the shape of the residual-free bubble as a benchmark against which we design the polynomial bubbles.

Table 1Edge bubble functions employed for fine-scale fields.

Element Bubble function

T3 4n 1� n� gð ÞQ4 1

2 1� n2� �

1� gð ÞT6 4n2 1� n� gð Þ2

Q9 14 1� n4� �

1� gð Þ2 þ 14 1� n2� �

1� gð Þ



As a representative study, we consider a right-triangular sector xðaÞs with unit width and a height r, as shown in Fig. 5(a); arepresentative submesh for computing the residual-free bubble is shown in the background by the dotted lines. The interfacesegment cs is taken as the base of the triangle, and the material tensor is specified as AðaÞ ¼ I, the second-order identity ten-sor. For various values of the height r, the stabilization parameter s � sðaÞs is computed using the polynomial bubble fromTable 1 as well as the residual-free bubble obtained from the solution of (70), and the results are shown on a log–log scalein Fig. 5(b). We observe that when the height or directed aspect ratio r 6 1, the polynomial (VMS) and residual-free (RFB)bubbles exhibit similar values and trends. However, when r > 1, the value of s for the polynomial bubble decreases whilethe value from the residual-free bubble remains nearly constant or slightly increases. This behavior is a result of the local-ization of the residual-free bubble toward the interface as the sector becomes increasingly slender. The shape of the bubblecomputed from (70) on a 64 � 64 grid is presented in Fig. 6. Clearly, the bubble is localized to the interface segment. Becausethe bubble functions in Table 1 are supported over the entire sector, they are unable to capture this localized behavior andthus produce an artificially low estimate for the stabilization parameter. Since the penalty term is inversely related to thevalue of s, the polynomial bubble approximation yields disproportionately large penalty parameters for this range of direc-ted aspect ratios r (see Fig. 5(a)) that in turn pollutes the numerical results at the interface. A study of the optimal range forthe stabilization parameter in the case of the Poisson equation is presented in [14]. The numerical error was found to be sig-nificant when the parameter was either too small, leading to loss of coercivity, or too large, leading to over-constraint and ill-conditioning. While the values of s for the polynomial bubbles falls close to the lower end of this optimal spectrum for smalldirected aspect ratio sectors, the value approaches the ill-conditioned higher end for large directed aspect ratios. We remarkthat while the solutions of (70) for r 6 1 are computed with the standard 64 submesh containing 64 � 64 cells, the values forr > 1 required submeshes with 256 cells along each edge to adequately resolve the features.

Fig. 6. Residual-free bubble for a sector with aspect ratio r ¼ 4 computed on a 64 � 64 submesh.

Fig. 5. Stabilization parametric study: (a) right-triangular sector xðaÞs with example submesh; (b) stabilization parameter s as a function of height r obtainedfrom various approximations.



Remark. As a reference, we have included a curve denoted by ‘‘Nit.’’ that represents the stabilization parameter associatedwith the standard Nitsche or DG method. This value is computed by assuming two identical sectors across the interface andapplying the formula for the Nitsche penalty parameter defined in Section 5.1 along with the formula for ss associated with

(41), which simplifies to s ¼ 12 meas xðaÞs

� �=meas csð Þ.

In order to improve the representation of the fine scales for large aspect ratios while avoiding the prohibitive expense ofcomputing the residual-free bubbles with high fidelity, we propose a modified algorithmic procedure for generating the bub-ble functions in such cases. The idea of adjusting low-order polynomial bubbles to approximate the character of residual-freebubbles has met with success in previous studies. One of the earlier examples includes the work of Brezzi et al. [10] whichproposed moving the internal node of the hat-type bubble functions to yield robust numerical performance at minimal ex-pense across the various convection–diffusion-reaction regimes for the one-dimensional case. Another example is the stabi-lized formulation for the Navier–Stokes equations proposed by Masud and Calderer [25] in which the fine-scale bubblefunction is selected in each element according to the most upwinding node in the coarse-scale flow features. In the samevein of these studies, we develop a procedure for reducing the area of non-zero support for the polynomial bubbles for highaspect ratio sectors, which is depicted in Fig. 7 for both triangular and quadrilateral elements. The physical coordinates of thesector vertices are denoted by the variables z while the parametric coordinates are denoted by the values within parenthe-ses. The thin blue dashed lines denote the template sector formed according to the steps outlined in Section 4.1. The key ideais that when the depth of the sector exceeds the width of the interface segment, a line parallel to the base (in parametricspace) is inserted to cut the sector into a region with edge lengths comparable to the segment width. For triangular elements,the midpoint of the line segment is adopted as the new vertex of the sector. This procedure is described more precisely inBoxs 1 and 2. In subsequent discussions, we will refer to the new sector formed by this procedure as a truncated sector. Weremark that the definition of the truncated sector collapses to the parent sector when the aspect ratio r ! 1 so that a sharptransition is avoided.

Box 1. Algorithm for defining truncated triangular sector

1. Compute l12 and l31; l32f g where lab ¼ za � zbk k2. If l12 > l3c where l3c ¼min l31; l32ð Þ:

Exit3. Else: Define reference point �z for cutting line:

�z ¼ zc þ l12=l3cð Þ z3 � zcð Þ

4. Find parametric location of �z ¼ �z �n; �g� �

5. Set �z3 ¼ z 12� 1

2�g; �g

� �6. Define sector xs by points z1; z2; �z3f g

Fig. 7. Definition of truncated sector: (a) triangular element; (b) quadrilateral element.



Box 2. Algorithm for defining truncated quadrilateral sector

1. Compute l12 and l32; l41f g where lab ¼ za � zbk k2. If l12 > ldc where ldc ¼min l32; l41ð Þ:


�z ¼ zc þ l12=ldcð Þ zd � zcð Þ

4. Find parametric location of �z ¼ �z �n; �g� �

5. Set �z3 ¼ z 1; �gð Þ and �z4 ¼ z �1; �gð Þ6. Define sector xs by points z1; z2; �z3; �z4f g

The preceding algorithm has a natural analog in three dimensions which is depicted in Fig. 8. Presently, the sector xðaÞs iscut by a plane parallel to the base at a parametric depth �f such that the physical depth is comparable to the longest edge ofthe triangular interface segment. The intersection of the plane with the original sector serves as the new top base of the trun-cated sector for hexahedral elements while the centroid of this triangle is used as the vertex of the truncated sector for tet-rahedral elements. This algorithm is presented in detail in Box 3 and Box 4.

Box 3. Algorithm for defining truncated tetrahedral sector

1. Compute l12; l23; l31f g and l41; l42; l43f g where lab ¼ za � zbk k2. If lb > l4c where lb ¼max l12; l23; l31ð Þ and l4c ¼min l41; l42; l43ð Þ:

Exit3. Else: Define reference point �z for cutting plane:

�z ¼ zc þ lb=l4cð Þ z4 � zcð Þ

4. Find parametric location of �z ¼ �z �n; �g;�f� �

5. Set �z4 ¼ z 13� 1

3�f; 1

3� 13�f;�f

� �6. Define sector xs by points z1; z2; z3; �z4f g

Box 4. Algorithm for defining truncated hexahedral sector

1. Compute l12; l23; l31f g and l41; l52; l63f g where lab ¼ za � zbk k2. If lb > ldc where lb ¼max l12; l23; l31ð Þ and ldc ¼min l41; l52; l63ð Þ:


�z ¼ zc þ lb=ldcð Þ zd � zcð Þ

4. Find parametric location of �z ¼ �z �n; �g;�f� �

5. Set �z4 ¼ z 0;0;�f� �

; �z5 ¼ z 1; 0;�f� �

, and �z6 ¼ z 0;1;�f� �

6. Define sector xs by points z1; z2; z3; �z4; �z5; �z6f g

The results from the two-dimensional analysis from calculating the stabilization parameter s using the modified, trun-cated sectors are shown as the green curve with superposed ‘‘x’’ denoted ‘‘VMSm’’ in Fig. 5(b). This curve agrees much moreclosely with the trend of the residual-free bubble. Therefore, we opt to employ the truncation procedure for the numericalstudies in Section 5.

Remark. The above approach for defining the modified sectors is influenced by the observed characteristics of the residual-free bubbles for this linear problem. Similar results were also obtained for a study of isotropic elasticity. Indeed, thetruncation method produced high quality results for the problems studied in Section 5. However, for more involved



problems incorporating anisotropy or nonlinearity, a more sophisticated procedure may be needed to more closelyapproximate the fine-scale field and thereby improve the accuracy of the computed solution. Generalizing and verifying thedesign of the bubble function for such cases will be addressed in future work.

5. Numerical results

In this section, we compare the performance of the proposed interface formulation and the associated fine-scale modelswith the standard Nitsche method for a series of four problems posed across a range of PDEs. The effects of nonmatchingmeshes and dissimilar material properties will be highlighted. Full numerical quadrature was used to evaluate all domainintegrals, and the three-point Gauss rule was used for boundary and interface integrals. To provide the optimal comparisonwith the Nitsche method, the following definition for the mesh size parameter h is used in the subsequent formulas for pen-alty terms:

h ¼ 2 meas csð Þ=meas Xð1Þes

� �þmeas csð Þ=meas Xð2Þes

� �h i.ð71Þ

where XðaÞesis the entire element adjoining segment cs. This definition has been found to yield relatively optimal numerical

performance [2,23].Throughout the examples that follow, we will assess the solution at the interface using two definitions of the numerical

flux. The first is called the ‘‘total flux’’ and is computed by k ¼ Auh� �

� sssuht, while the second is called the ‘‘gradient flux’’and is computed by ~k ¼ Auh

� �. The total flux is analogous to the standard definition of the numerical flux used in Discon-

tinuous Galerkin methods where the jump term is included. However, the gradient flux can be used to assess the qualityof the gradient of the solution in the elements adjacent to the interface and to discern how much the penalty term contrib-utes to the total flux. For the Nitsche method, the same formulas are applied except that the weights are taken asdð1Þ ¼ dð2Þ ¼ 1

2 and the penalty parameter is computed as a function of h given by (71), where the appropriate functional formfor the corresponding PDE is presented in the descriptions of the problems in the following sections.

5.1. Poisson problem

The first problem involves the analysis of a Poisson problem with an analytical solution. A rectangular domain with tworegions containing different coefficient matrices Að1Þ ¼ Að1ÞI and Að2Þ ¼ Að2ÞI are considered as shown in Fig. 9. The sourceterm in each region is prescribed such that the exact solution is given by:

uð1Þ ¼ sin pxð Þ sin pyð Þ ð72Þ

uð2Þ ¼ sin pxð Þ sin pyð Þ þ p Að1Þ � Að2Þ� �

=Að2Þh i

sin pxð Þ cos14p

� �y� 1

4

� �ð73Þ

This problem was designed such that the solution and the flux remain continuous across CI when Að1Þ – Að2Þ. Two investiga-tions were performed to compare the quality of solutions obtained from the proposed interface method and the standard

Nitsche method as the ratio of material properties is varied from Að2Þ=Að1Þ ¼ 10�6 to Að2Þ=Að1Þ ¼ 10þ6. In the first case, a uni-

Fig. 8. Definition of truncated sector: (a) tetrahedral element; (b) hexahedral element.



form discretization is applied such that the interface is conforming, with 17 nodes along the horizontal direction in both re-

gions and 5 and 13 nodes in the vertical direction in regions Xð1Þ and Xð2Þ, respectively. Both linear and quadratic triangularand quadrilateral elements were employed with the diagonals for the T3 elements oriented upper-right to lower-left andupper-left to lower-right for the T6 elements. In the second case, a nonconforming interface is generated by increasing

the number of nodes along the horizontal in the region Xð1Þ to 41. For the Poisson equation, the Nitsche penalty parameter

is specified as ss ¼ max Að1Þ;Að2Þ� �

�=h, and we take � ¼ 1.

For each value of the material ratio Að2Þ=Að1Þ, the L1 norm of the error in the two flux measures e ¼ t � k and e ¼ t � ~k wascomputed for both methods, where the exact flux is given by t ¼ Að1Þp sin pxð Þ cos 1

4 p� �

with a maximum value oftmax ¼ Að1Þp cos 1

4 p� �

. The relative error, e=tmax, is presented on a log–log scale for all element types and both numerical meth-ods in Fig. 10(a) and (b). For the case of equal material properties, both methods exhibit approximately the same level oferror in the total flux. However, the proposed interface method produces results with uniform levels of error as the materialratio is varied. In contrast, the Nitsche method yielded results with errors in excess of 100% for all element types whenAð2Þ=Að1Þ > 100, as clearly seen in the plot of the gradient flux in Fig. 10(b). While the jump term compensates for this errorin the quadrilateral elements to produce lower error for the total flux, the triangular elements exhibit poor performance all-around. We remark that both methods perform well for the case Að2Þ=Að1Þ < 1 because the gradient becomes essentially con-stant in the y-direction in Xð2Þ and thereby making the problem easier to solve.

The discrepancy in the performance of the two methods can be explained by comparing the magnitude of the penaltyparameter associated with each material ratio, which is plotted in Fig. 10(c). The value of the penalty parameter ss forthe VMS interface formulation decreases when the material properties are varied because the weighting terms dðaÞs in thenumerical flux can compensate for the imbalance and place more emphasis on the side of the interface with a lower materialcoefficient. However, the Nitsche method does not offer this flexibility and instead must increase the penalty term to main-tain stability at the expense of numerical accuracy.

Similar trends were observed for the flux errors for the nonconforming interface, which are presented in Fig. 11. In each

element type and interface formulation, the error for equal material properties Að2Þ=Að1Þ ¼ 1� �

was higher for the noncon-

forming mesh compared to the conforming mesh except for the linear triangles. We also note that the maximum error in

the total flux is higher for the proposed interface method compared to the Nitsche method for the case Að2Þ=Að1Þ ¼ 1. How-ever, the error in the gradient flux is consistently lower.

To further investigate the behavior of the methods for nonconforming interfaces, we consider the particular mesh of lin-ear triangles shown in the contour plots of Fig. 12 and fix the material properties Að1Þ ¼ Að2Þ ¼ 1; the perspective is zoomed inover the lower-center portion of the domain. For this mesh, the size of the elements in both regions is identical, but theirorientation with respect to the interface is different. From the contour plot of the y�component of the material gradientAðaÞ � ruðaÞ in Fig. 12(a), we see that the solution from the proposed method retains the symmetry of the exact solution moreclosely than the Nitsche solution in Fig. 12(b). While the contours far from the interface are identical for both methods, theNitsche results exhibit a higher level of discontinuity between the two regions.

Next, we present a line plot of the total flux and gradient flux for both methods in Fig. 13. The blue curve representing thegradient flux in 13(b) confirms the behavior in Fig. 12(b), namely that the average of the gradients from the two regions hasdrifted away from the exact solution and is no longer symmetric about x ¼ 1. However, the gradient flux for the VMS inter-face method, which is computed from a weighted average, closely agrees with the exact flux. For both methods the total fluxcurves fluctuate about the exact curve, although the amplitude is higher for the VMS results. This behavior can be partiallyattributed to the larger value of the penalty parameter computed by the proposed method, which is about four times thevalue computed from the Nitsche approach. The results for the fluxes from the Nitsche method with � ¼ 4 are shown inFig. 13(c). We observe that the gradient flux still exhibits more error than the VMS results, and that the total flux now con-tains fluctuations of the same magnitude as in Fig. 13(a).

Remark. From these investigations, we conclude that the proposed interface method produces numerical results that arerobust with respect to material mismatch and nonconforming interfaces.

Fig. 9. Problem description for Poisson problem with two material regions.



5.2. Beam bending problem

The next problem is an elastic beam under pure bending with an interface at the midsection that separates it into regionswith different isotropic material properties. Considering plane stress conditions and placing the origin of coordinates at theintersection of the centerline and middle-plane, the solution in each half of the beam is given by:

uðaÞx ¼2Mxy

EðaÞDð74Þ

uðaÞy ¼ �M x2 � my2� �

EðaÞDð75Þ

Fig. 10. Relative flux errors versus material ratio, conforming mesh: (a) total flux; (b) gradient flux; (c) value of penalty parameter ss .



For our analysis, we take the length of the beam to be L ¼ 16, the full depth D ¼ 4, and the end moments as M ¼ 72; addi-tionally, we set mð1Þ ¼ mð2Þ ¼ 0 to admit a continuous solution along the entire length. This problem has served as a bench-mark for embedded mesh and embedded interface techniques [34,1], where mismatch in the material and/ordiscretization can lead to stress locking or pollution of the solution in the elements close to the interface. Similarly, we placean interface at the junction of the two halves of the beam that are meshed with linear triangular elements and investigate thequality of the interface tractions as well as the surrounding stress field obtained from the proposed interface method. In thesimulations that follow, only the top half of the beam was modeled, and anti-symmetry conditions were applied along thecenterline. Also, the exact displacement field given by (74) and (75) is applied as boundary conditions to the ends of the

beam. For linear elasticity, the Nitsche penalty parameter is taken as ss ¼ max Gð1Þ;Gð2Þ� �

�=hh i

I, where G is the shear

modulus.

Fig. 11. Relative flux errors versus material ratio, nonconforming mesh: (a) total flux; (b) gradient flux; (c) value of penalty parameter ss .



As the first example, we set the Young’s modulus to Eð1Þ ¼ 102 and Eð2Þ ¼ 107 (Xð1Þ is the left half of the beam) and employa conforming discretization with 32 (�2) elements horizontally and 16 elements vertically in each domain. The normal inter-face traction obtained from the stabilized interface method and the Nitsche method with � ¼ 1 are shown in Fig. 14(a) and(b), respectively. In both plots, the constant-strain triangles yield a stair-step curve for the total flux centered around theexact solution. However, fluctuations are evident at the top surface of the beam of the Nitsche results; additionally, the gra-dient flux for the VMS interface method is not centered around the exact solution. We compare this behavior to the contourplots of rxx for both methods shown in Fig. 15. For the results from the proposed method, the stress pattern is uninterruptedbetween the two regions such that the interface (located at the center) is invisible. However, slight disturbances are visiblefor the stress results from the Nitsche method. Thus, the extremely high value for ss from the Nitsche method, which is fourmagnitudes greater than from the VMS method for this case, leads to a slight degradation of the results. In contrast, the pro-posed interface formulation yields values of dð1Þ � I and dð2Þ � 0 for the flux weight parameters according to (50), whichemphasizes the more flexible left half of the beam. In fact, comparing Figs. 14(a) and 15(a), we observe that the gradient flux

Fig. 12. Contour plot of AðaÞ � ruðaÞ� �

y: (a) stabilized interface method; (b) Nitsche method.

Fig. 13. Interface flux: (a) stabilized interface method; (b) Nitsche method, � ¼ 1; (c) Nitsche method, � ¼ 4.



n � rh� �

agrees precisely with the value of the stress from the elements on the left side of the interface, which each have aslightly lower value than the corresponding element on the right half.

For the second analysis, we select equal material properties Eð1Þ ¼ Eð2Þ ¼ 103 but generate the nonconforming mesh shownin Fig. 16. Also, for this case we increased the value of the penalty coefficient for the Nitsche method � ¼ 4 because the meth-od produced unstable results for � ¼ 1, as indicated in Fig. 17; the dependence of the numerical stability on this user-definedparameter is one drawback of the Nitsche method. Similar to the results for the Poisson problem in Section 5.1, we see thatthe difference in element orientation causes more disturbances in the stress pattern near the interface compared to the sur-rounding bulk domain for the Nitsche method than for the proposed interface method, although the difference is relativelysmall. From the interface traction plots in Fig. 18(b), we observe that the Nitsche results for the gradient flux are slightlyoffset from the exact traction curve. In comparison, the total flux curve from the proposed interface method lies directlyon top of the true solution with nearly constant step sizes. Finally, we also provide curves for the shearing traction

Fig. 14. Interface normal traction: (a) stabilized interface method; (b) Nitsche method.

Fig. 15. Stress rxx contour plots: (a) stabilized interface method; (b) Nitsche method.




Fig. 18(c) and (d) and remark that both methods produce values that are close to zero. The values from the proposed inter-face method exhibit slightly higher variation than from the Nitsche method.

As a third case for the beam problem, we employ a nonconforming and unstructured discretization for both halves of thebeam such that the mesh on one side of the interface is more refined and the material properties are set to Eð1Þ ¼ 107 andEð2Þ ¼ 102. Also, the value of the Nitsche parameter is reset to � ¼ 1. Corresponding plots of the interface traction and stresscontours for both methods are shown in Figs. 19 and 20, respectively. The overall quality of the solutions seems comparablebetween the two methods for this example. While the fluxes for the proposed method are biased toward the right domain

Fig. 17. Interface normal traction for Nitsche method, � ¼ 1.

Fig. 18. Interface tractions: (a) normal, stabilized interface method; (b) normal, Nitsche method; (c) shear, stabilized interface method; (d) shear, Nitschemethod.



and are relatively smooth except for a few sharp features, the Nitsche results are rougher but remain closer to the exact trac-tion curve. Also, the stress contours for the Nitsche method exhibit disturbances near the endpoints of the interface (y ¼ 0and y ¼ 2Þ. The interplay of smaller element size with larger material stiffness could help balance the Nitsche results. None-theless, these plots illustrate that the automatic procedure for computing the penalty and weighting parameters in the pro-posed interface method yields stable solutions for unstructured grids.

5.3. Thermal beam

Next, we consider a thin beam subjected to a slowly increasing heat flux q tð Þ on its top surface, as shown in Fig. 21. Theother surfaces of the beam are taken to be perfectly insulated q ¼ 0ð Þ. Under plane stress conditions and neglecting rate ef-fects, the solution for the displacement field and nonzero stress fields in the beam resulting from the thermal effects are ob-tained as described in [5]:


Fig. 20. Interface normal traction: (a) stabilized interface method; (b) Nitsche method.

Fig. 21. Problem description for beam under surface heat flux.



ux ¼xbE

NT

A� zMT q tð Þ

I

ð76Þ

uz ¼ �x2bMT q tð Þ

EI� mb

ENT z

A�MT q tð Þz2

2I

þ a 1� mð Þ

Z z

0T fð Þdf ð77Þ

rxx ¼ ryy ¼aE

1� mq tð Þ4hk

h2

3� z2

!ð78Þ

NT ¼aEqc

Z t

0q sð Þds; MT ¼

aEq tð Þh3

3kð79Þ

Z z

0T fð Þdf ¼ q tð Þ

12khz3 þ q tð Þ

4kz2 þ 1

2hqc

Z t

0q sð Þds� hq tð Þ

12k

z ð80Þ

A ¼ 2bh; I ¼ 23

bh3 ð81Þ

where E; m; q are mechanical material properties and a; k; c are thermal material properties. As a simple example, we takethe dimensions of the beam to be L ¼ 4; b ¼ 1; 2h ¼ 1 and adopt the following values for the parameters:

E ¼ 1000; m ¼ 0:25; q ¼ a ¼ k ¼ c ¼ 1 ð82Þ

Finally, the heat flux is specified as the unit ramp function q tð Þ ¼ t, and we perform the simulation at time t ¼ 1 such thatR t0 q sð Þds ¼ 1

2. Only the right half of the beam is modeled, and the stress field from the exact solution is applied as tractions onthe vertical faces.

Our objective for analyzing this problem is to assess the consistency of the additional term present in the formulation forresidual stresses (56). Thus, although the physical problem does not contain an interface, we employ a nonconforming meshof T6 elements with a slanted interface as shown in Fig. 22. First, we compute the solution using the weak form (56) retainingthe interface residual stress term, and then the problem is solved again without including the extra term. The contour plot ofthe unprocessed rxx stress field from both simulations is given in Fig. 22. The numerical solution is quite smooth for the con-sistent version in Fig. 22(a) and closely approximates the actual solution. However, the result obtained from neglecting theresidual stress term exhibits significant fluctuations at the interface. Thus, we conclude that this term is a necessary com-ponent of the stabilized formulation in the presence of thermal strains. Additionally, this analysis demonstrates the utilityof deriving the numerical flux for problems in which the expression for the complete weak form may not be readily apparent.

5.4. Simply-supported beam: multi-PDE benchmark study

As a benchmark problem for the multi-PDE case, we consider the problem of a simply-supported elastic beam with a bodyforce representing its self-weight. The problem description is shown in Fig. 23, where plane strain conditions are assumedthroughout. The exact displacement and pressure fields for this problem are referenced in [29] along with the expression forthe consistent traction fields that are prescribed on the left and right edges. Also, we adopt the following geometric andmaterial parameters: C ¼ 1; L ¼ 5; E ¼ 7:5� 107; m ¼ 0:3; q ¼ 1000, and g ¼ 9:81.

For our computational model of this problem, we treat the center of the beam using the mixed elasticity formulation ofSection 3.4 and the outer regions using the pure-displacement formulation of Section 3.2. Thus, the composite formulas fromSection 3.5 are utilized for the numerical flux and stability parameters. A convergence rate study is conducted, whereby eachregion is discretized using different polynomial orders or element shapes such that the interfaces are truly nonconforming.Details of the mesh hierarchy are contained in Table 2, which corresponds to a study by Masud et al. [29] in which a mixed

Fig. 22. Stress rxx contour plot: (a) including residual stress term; (b) neglecting residual stress term.



elasticity formulation is used throughout and the beam is treated as incompressible. In the example discretizations providedin Fig. 24, the quadratic elements are grouped in the center of the beam coinciding with the mixed elasticity portion, as indi-cated by the dashed-red zone in the center of Fig. 23. These disparate combinations of PDEs and element types represent astress test for the interface formulation.

The convergence of the displacement error measured in the L2 norm and H1 seminorm are presented in Fig. 25; the valueshave been normalized with respect to the norms of the exact solution. In all cases, the rates conform with the optimal ratespredicted from finite element theory. Namely, the meshes containing linear elements exhibit a rate of 2.0 in the L2 normwhile the T6–Q9 meshes achieve the rate of 3.0. Similar trends are evident in the H1 seminorm error plots. Thus, we concludethat the composite interface formulation produces stable and convergent results for the coupling of different governingequations.

As a qualitative assessment of the solution accuracy provided by the interface formulation, in Fig. 26 we compare the con-vergence of the centerline deflection on the bottom chord, at the point x; yð Þ ¼ 0;�Cð Þ, obtained from the T3–Q4 mesheswhen different combinations of the constitutive models are employed. The naming convention is as follows: Q4(u)–Q9(u)corresponds to Section 3.2, Q4(p)–Q9(p) corresponds to Section 3.4, and Q4(p)–Q9(u) corresponds to Section 3.5. As wouldbe expected, the pure-displacement formulation provides the stiffest approximation of the beam and thus exhibits the larg-est error, while the mixed elasticity formulation in general provides the lower error. In comparison, the multi-PDE resultsexhibit a steady convergence that is more accurate than the pure-displacement method and comparable to the mixed meth-od. Finally, we also provide a contour plot in Fig. 27 of the unprocessed stress field rxx obtained on the coarse Q4–Q9 mesh.No smoothing techniques have been applied to the field; the strains and stresses are directly evaluated through differenti-ating the finite element displacements and evaluating the constitutive relations (44) and (57), respectively. The quality ofthis contour plot is quite striking. We also remark that the weighting tensors for the numerical flux across the interfacefor this discretization are computed as matrices with values dðQ9Þ ¼ 0:73 0; 0 0:69½ � and dðQ4Þ ¼ 0:27 0; 0 0:31½ �for the Q9 and Q4 element regions, respectively. Thus, the simple average definition for the numerical flux may be a poorapproximation in the present case.

Table 2Listing of the number of elements and nodes in the mesh hierarchy.

Mesh name Triangular (T3–T6) Quadrilateral (Q4–Q9) Linear (T3–D4) Quadratic (T6–Q9)

Elements Nodes Elements Nodes Elements Nodes Elements Nodes

Coarse 40 55 20 55 28 42 28 125Medium 160 171 80 171 112 125 112 176Fine 640 595 320 595 448 423 448 1547Very fine 2560 2211 1280 2211 1792 1547 1792 5907

Fig. 23. Simply-supported beam problem description.

Fig. 24. Uniform mesh hierarchy: (a) 40 element mesh, mixed order triangular; (b) 20 element mesh, mixed order quadrilateral; (c) 28 element mesh,mixed type.



6. Conclusion

We have presented a novel derivation of the classical Discontinuous Galerkin (DG) method from an underlying Lagrangemultiplier formulation through a Variational Multiscale (VMS) approach. By modeling the numerical fine scales locally atdiscrete interfaces that arise due to nonconforming meshes, material mismatch, disparate governing PDEs, or unstable pri-mal/multiplier interpolation combinations, we obtain residual-driven terms that stabilize the Lagrange multiplier method.Adopting a discontinuous functional space for the multipliers, an analytical expression for the multiplier field along interfacesegments is derived, which serves as the definition for the numerical flux. Upon substitution back into the stabilized mixedinterface formulation, a variational form is obtained that is form equivalent to the classical DG or Nitsche method in whichthe primary field is the only unknown. This generalized framework provides a consistent and stable platform for the couplingof different element types, different material properties, and even different governing equations across discrete nonconform-ing interfaces. Crucially, definitions for the penalty parameter and a weighted average numerical flux arise naturally duringthe course of the derivation that account for the element size and geometry as well as the variation in material properties oneach side of the interface.

Fig. 25. Convergence rates of normalized standard error: (a) L2 norm of displacement; (b) H1 seminorm of displacement.

Fig. 26. Convergence of normalized bottom center-point displacement.

Fig. 27. Contour plot of unsmoothed stress rxx on coarse Q4–Q9 mesh.



To ensure an efficient numerical implementation, we adopted a simple representation of the localized fine-scale fieldsalong the interface via polynomial bubble functions generated through an automatic procedure. An analysis was conductedcomparing the stabilization parameters computed using either the polynomial bubbles or residual-free bubbles for the Pois-son equation, and similar trends were observed when a modification for sectors with high aspect ratios was incorporated.Subsequently, representative numerical problems were studied that involved different element types and nonconformingmeshes for the Poisson equation, linear elasticity, and coupled pure-displacement and mixed elasticity. The proposed inter-face method is compared against the standard Nitsche method for the problems using single PDEs, and we observed that thepenalty parameter and weighted flux derived from the VMS approach exhibited robust performance. In particular, accuratenumerical results for the interfacial flux were obtained that did not pollute the accuracy of the solution in the surroundingneighborhood of the interface. These same attributes carried over to the multi-PDE benchmark elasticity problem. In the fu-ture, we aim to extend the stabilized interface formulation to nonlinear PDEs for which analytical expressions for the sta-bilization parameter as a function of the evolving solution field would be a tremendous asset.

Acknowledgments

This work was sponsored by an NSF Graduate Research Fellowship. This support is gratefully acknowledged. Authorswould like to thank Professor I. Harari for helpful discussions.

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