DPG method with optimal test functions for a transmission problem Supported by CONICYT through FONDECYT projects 1110324, 3140614 and Anillo ACT1118 (ANANUM).

# DPG method with optimal test functions for a transmission problem Supported by CONICYT through FONDECYT projects 1110324, 3140614 and Anillo ACT1118 (ANANUM).

## Abstract

We propose and analyze a numerical method to solve an elliptic transmission problem in full space. The method consists of a variational formulation involving standard boundary integral operators on the coupling interface and an ultra-weak formulation in the interior. To guarantee the discrete inf-sup condition, the system is discretized by the DPG method with optimal test functions. We prove that principal unknowns are approximated quasi-optimally. Numerical experiments for problems with smooth and /media/arxiv_projects/18759/singular solutions confirm optimal convergence orders.

: transmission problem, coupling, DPG method with optimal test functions, ultra-weak formulation, boundary elements, finite elements

: 65N30, 35J20, 65N38

## 1Introduction

In recent years, Demkowicz and Gopalakrishnan have established the discontinuous Petrov-Galerkin method with optimal test functions as a method that is designed to be stable [17]. In particular, it aims at robust discretizations of /media/arxiv_projects/18759/singularly perturbed problems, cf. [10]. In general terms, this method consists in applying Petrov-Galerkin approximations with optimal test functions to (in most cases) ultra-weak variational formulations (see [13] for an early use of ultra-weak formulations). Such a variational formulation is obtained by element-wise integration by parts on some partition and the replacement of appearing terms on the elements’ boundaries with new unknowns (see [5] for the idea of introducing independent boundary unknowns).

Until recently the DPG method with optimal test functions has been studied only for problems on bounded domains. In [24], we considered boundary value and screen problems of Neumann type which can be reduced to a hyper/media/arxiv_projects/18759/singular boundary integral equation. This includes the case of a PDE on an unbounded domain. In this paper we study for the first time a DPG strategy with optimal test functions for a transmission problem. This problem is of academic nature (Poisson equation) and set in the full space. We expect that our fairly general approach is applicable to more practical cases of transmission problems with /media/arxiv_projects/18759/singular perturbations on bounded subdomains. This is feasible whenever there is a DPG finite element technique available for the corresponding /media/arxiv_projects/18759/singularly perturbed problem on the subdomain.

Transmission problems often appear in the modeling of multiple physical phenomena, and therefore require the combination of two, possibly different, numerical methods. A popular approach is the coupling of finite elements and boundary elements. Whereas finite elements can be used in a relatively straightforward way to solve nonlinear PDEs with space-dependent coefficients and sources, it is a challenging task to apply them on unbounded domains. On the other hand, boundary elements can deal naturally with unbounded domains but, by construction, work best for linear, homogeneous PDEs with constant coefficients.

There are different approaches for the coupling of finite and boundary elements. In this paper we consider the so-called nonsymmetric or one-equation coupling, also referred to as Johnson-Nédélec coupling. It has the practical advantage of involving only two of the four classical boundary integral operators. This coupling was mathematically analyzed first in [8]. The mathematical proofs in the mentioned works require one of the involved boundary integral operators (the double-layer operator) to be compact. In case of the Laplace equation this property does not hold on polyhedral domains and, in the case of linear elasticity, not even on domains with smooth boundary. In [30], Sayas proved the well-posedness of the Johnson-Nédélec coupling without u/media/arxiv_projects/18759/sing compactness arguments and thus gave a mathematical justification of the nonsymmetric coupling even on polyhedral domains. Since then, different authors have re-considered nonsymmetric couplings, see [1]. For an extensive discussion of this topic, we refer to [20].

In this paper, we extend the nonsymmetric coupling of Johnson-Nédélec to a DPG method with optimal test functions. More specifically, given a coupled system of PDEs (one in a bounded domain, one in the exterior), we use an ultra-weak formulation for the interior part coupled to classical boundary integral equations for the exterior problem. The whole system is discretized by the Petrov-Galerkin method with optimal test functions.

An alternative approach would be to couple standard boundary elements with a DPG scheme restricted to the interior problem, i.e., optimal test functions are only used for the interior (finite element) part. In this case, however, several difficulties arise. For instance, it is unclear how to choose the remaining test functions to generate a square system. The design and analysis of a DPG-BEM coupling is left for future research. In contrast, the approach of computing optimal test functions for the whole system (as in this paper) appears to be the most generic one and deserves a thorough analysis.

An outline of this paper is as follows. In Section 2, we introduce the model problem and present the mathematical framework. We also summarize some results related to the DPG method, to be used subsequently. In Subsection Section 2.4 we formulate the method and state the main results. Theorem ? establishes stability of the continuous variational formulation and Theorem ? shows the quasi-optimality of conforming discretizations (so-called Céa-lemma). Proofs of the two theorems are given in Section 4. Principal part of these proofs involves the analysis of adjoint problems. This analysis is made in Section 3. In Section 5 we present some numerical results that underline our theory. Some conclusions are drawn in Section 6.

## 2Mathematical setting and main results

Let , , be a bounded Lipschitz domain with boundary . Our model transmission problem is as follows: given volume data and jumps , , find and such that

The normal vector on points in direction of . Here, , , , and the spaces are of standard Sobolev type (some more details are given in the next section). For , we assume that and . The scaling condition on is to ensure the ellipticity of the /media/arxiv_projects/18759/single layer operator, and the compatibility condition on the data and is needed in order to use the radiation condition in the form .

### 2.1Abstract DPG method

We briefly recall the abstract framework of the DPG method with optimal test functions, cf. [16]. We state this in a form that will be convenient for the forthcoming analysis. The continuous framework is provided by the following result which is a consequence of the open mapping theorem and the properties of conjugate operators, cf. [33]. In this manuscript, all suprema are taken over the indicated sets except .

For a proof of we refer to [34]. Now suppose additionally that is a Hilbert space. Given a bilinear form , and a linear functional , we aim to

For an approximation space , the Petrov-Galerkin method with optimal test functions is to

Here, is the trial-to-test operator defined by

and is the inner product in which induces the norm . The following result is a consequence of the Babuška-Brezzi theory [2]. For a proof of the best approximation property see [17].

### 2.2Sobolev spaces

We use the standard Sobolev spaces , , , , for Lipschitz domains . Vector-valued spaces and their elements will be denoted by bold symbols. In addition, we use spaces on the boundaries of Lipschitz domains . Denoting by the trace operator, we define

and equip them with the canonical trace norm and dual norm, respectively. Here, duality is understood with respect to the extended inner product . The inner product will be denoted by . Let denote a disjoint partition of into open Lipschitz sets , i.e., . The set of all boundaries of all elements is the skeleton . By we mean the outer normal vector on for a Lipschitz set . On a partition, we use product spaces and , equipped with respective product norms. The symbols and denote the -piecewise gradient and divergence operators. We use spaces on the skeleton of , namely

These spaces are equipped with the norms

Note that we think of the skeleton not as one geometric object, but rather as the set of boundaries of all elements. Consequently, we have defined and not as canonical trace spaces but as product spaces of trace spaces. This subtle difference simplifies the subsequent analysis. For two functions and we use the notation

Note that for and integration by parts shows that

and so the above left-hand side makes sense also for functions and . We use an analogous duality pairing for functions and . For and we define norms of their jumps across by duality,

Here, on . We will need the following estimates.

The estimates – follow straightforwardly by integration by parts, e.g., cf. [16] for . The estimate follows by definition of the norms in and ,

The estimate follows the same way, u/media/arxiv_projects/18759/sing that is equivalently described as the space of normal components on of functions in , and that , cf. [22].

### 2.3Boundary integral operators

In order to incorporate the PDE given in the exterior domain , the classical boundary integral operators will be used. The fundamental solution

of the Laplacian gives rise to the two potential operators and defined by

Then, boundary integral operators are defined as (/media/arxiv_projects/18759/single layer operator) and (double layer operator) with adjoint . The operators , , , and are bounded, and there holds the representation formula

for solutions of the exterior PDE . We refer to [15] for proofs and more details /media/arxiv_projects/18759/regarding boundary integral equations and the above operators.

### 2.4Nonsymmetric coupling with ultra-weak formulation and main results

The trial space of our variational formulations will be . This space is a Hilbert space with norm

In addition, we will need the space , which is a Hilbert space with norm

Note that the canonical restrictions of and show that can be viewed as an element of . U/media/arxiv_projects/18759/sing this restriction, we can /media/arxiv_projects/18759/regard as a subspace of . However, is only a seminorm on . The test space of our formulation will be , being a Hilbert space with norm

In addition, we will need the space , which is a Hilbert space with norm

Note that . The variational formulation that we will analyze is the following Johnson-Nédélec type coupling: find such that

for appropriate test functions . The equations – are obtained by treating the interior PDE as in the DPG-finite element method, cf. [16], i.e., writing it as a first order system, testing with appropriate functions, integrating by parts piecewise, and replacing the appearing boundary terms by new unknowns and . These new unknowns already involve the interior trace and normal derivative of on , which are coupled to the exterior problem by u/media/arxiv_projects/18759/sing the interface conditions – in the representation formula . In contrast to that, in the classical nonsymmetric coupling, cf. below, the unknowns are and , where is the normal derivative of on .

The bilinear form on the left-hand side of will be called , and the linear form on the right-hand side will be called . We will use two different formulations which differ in the underlying spaces; the one we actually analyze and solve numerically is

The second one is only of theoretical interest and will be needed in the proofs of the main theorems, it is

The first result of this work states unique solvability and stability of the variational formulation . The proof will be given in Section 4 below.

The second main result of this work is the following quasi optimality result of the Petrov-Galerkin method with optimal test functions associated with the norm .

For the proofs of Theorems ? and ? we will apply the results of Lemmas ? and ?, hence we have to check the assumptions of Lemma ?. The boundedness of the bilinear form is shown in Lemma ?. The bijectivity of the operator that corresponds to the bilinear form will be proved in Section 4 below. In a first step, this will yield the results of Lemma ?, i.e., stability and quasi-optimality in the norm defined in Lemma ?. To obtain the results in the main theorems, it remains to relate the norm to the norms and . This will be done by characterizing the optimal test norms (optimal to ) and (optimal to ) and by relating them to the norm (optimal to ). These norm equivalences are the topic of Section 3.2.

## 3Technical results

We start by showing the boundedness of the bilinear form .

The proof of follows immediately by the Cauchy-Schwarz inequality and standard boundedness properties. Note that by definition of the norms and and and , it holds that

Furthermore, the boundedness of the operators , and , yield

For the proof of , note in addition that

due to the definition of the norm . The part is treated the same way.

### 3.1Johnson-Nédélec coupling

The aim of this subsection is to show that our new formulation is equivalent to the classical nonsymmetric coupling. As we will see in Section 4, this implies, in particular, injectivity of the operator that corresponds to the bilinear form .

The transmission problem can be written equivalently as: Given , find such that

for all . Proof of unique solvability is not straightforward as Problem is not elliptic, and was addressed recently in [30] and also in [1]. Following the approach of [1], the following stability result can be shown.

Denote the bilinear form on the left-hand side of as and the linear functional on the right-hand side as . Consider the problem of finding such that

for all . According to [1], a solution of also solves and vice versa. Furthermore, [1] states that the bilinear form on the left-hand side of is continuous and elliptic on . The norm of the linear functional on the right-hand side of is bounded by

We finish the proof by application of the Lax-Milgram lemma.

The next lemma shows that our new formulation is equivalent to the classical formulation .

We first show (i). U/media/arxiv_projects/18759/sing in shows that

Hence, and eventually . This yields . Identity and integration by parts implies for all . Integration by parts also shows for all . From and we conclude that

U/media/arxiv_projects/18759/sing the symmetry of , this leads us to

If we plug the last identity into , we obtain exactly for all . In total, is a solution of . Furthermore, it is also a solution of . This follows immediately as by the canonical restriction, and as .

Now we show (ii). To that end, denote by a solution of . Equation first shows that and hence , as well as . As , we have , and from follows . Likewise, follows from u/media/arxiv_projects/18759/sing . The case that is a solution of follows analogously.

We additionally need the following stronger result, which shows the surjectivity of the operator associated to our bilinear form.

U/media/arxiv_projects/18759/sing the Riesz representation theorem, we write

with and and . According to Lemma ?, there is a unique solution of with right-hand side data , , arbitrary and . From it follows that with . Now define , , , and . Integration by parts shows

Furthermore, shows that . Therefore, by definition of and , we conclude that . We have thus shown that for all .

### 3.2Bielak-MacCamy coupling and norm equivalences

In order to relate the norm to a norm of our choice, we will investigate norm equivalences in the test spaces. To that end define seminorms in and by

Here, is a bounded and linear extension operator, i.e., . See [22] for an explicit construction of . Equivalence of norms in the test space amounts to an analysis of the adjoint problem. In case of the nonsymmetric coupling, the adjoint problem is the so-called Bielak-MacCamy coupling, which first appeared in [3]. Given , it consists in finding such that

for all . Again, proof of the existence of a unique solution is not straightforward since the problem is not elliptic. However, u/media/arxiv_projects/18759/sing ideas from [1], the following result can be shown.

Denote the bilinear form on the left-hand side of as and the linear functional on the right-hand side as . Consider the problem of finding such that

for all . According to [1], a solution of also solves and vice versa. Furthermore, [1] states that the bilinear form on the left-hand side of is continuous and elliptic on . The norm of the linear functional on the right-hand side of is bounded by

Hence, by application of the Lax-Milgram lemma, the statement is proved.

We need the following extension of [16].

Choose as solution of the Bielak-MacCamy coupling with the respective data , and . By Lemma ?, such a solution exists uniquely and fulfills as well as

Now define , i.e., holds. At first, only, but testing with shows that with , i.e., . Then, testing with shows that

which gives . By definition of , there holds

and together with , this shows .

To see that a solution of is unique, assume that is a solution of with vanishing right-hand side. Then in and on . Therefore, in and hence . However, and also show that . We conclude that , which implies and .

We set in Lemma ? the right-hand side to be , , , and . Then, is the unique solution of , and the statement follows from .

The statement follows with the triangle inequality, the estimates –, and the continuity of the operators , , and .

## 4Proofs of main theorems

The next lemma shows that our new bilinear form is definite.

The implications “” in the statements are obvious. Suppose that for all . This means that solves with data and . Due to Lemma ? (ii), is a solution of with the respective data , , , and . We conclude that solves with all right-hand side data equal to . Lemma ? shows and . Lemma ? (ii) finally shows that .

Now suppose for all . Testing with shows , and testing with shows that and on . Furthermore, testing with appropriate shows on all , such that piecewise integration by parts yields

for all . Hence . Now, as in and on , it holds in . This implies that on it holds

where the last equation follows from the definition of . This shows that and hence . It follows that and .

With exactly the same reasoning one proves that is definite on .

We will use Lemma ? with , , and . First we check that the assumptions from Lemma ? hold. Clearly, and are reflexive Banach spaces with their respective norms. The operator is linear, bounded due to Lemma ?, injective due to Lemma ?, and surjective due to Lemma ?. Hence, Lemma ? is applicable. We obtain a unique solution with stability and best approximation in the norm . It remains to show the bounds

We start with the upper bound. Inspection shows that is the optimal test norm to . Hence, we can use Lemma ? and the left one of the identities (the assumptions in Lemma ? have been checked above) to conclude the upper bound in .

Now we show the lower bound in . Due to Lemma ? and we first obtain

Inspection shows that is the optimal test norm to . It remains to check that the left-hand side of is indeed . To that end, we will again apply Lemma ?, but this time with , , and . We check again that the assumptions hold. The spaces and are reflexive Banach spaces with their respective norms. The operator is linear and bounded due to Lemma ?. Bijectivity follows from the Babuška-Brezzi theory which applies by Lemmas ? and ?. Hence, by Lemma ?, the identity on the left of shows that the left-hand side of is indeed . This shows the lower bound in and concludes the proof of Theorems ? and ?.

## 5Numerical experiments

We conducted two numerical experiments for to support our analysis. As partitions we choose /media/arxiv_projects/18759/regular triangulations, i.e., all elements are triangles, and there are no hanging nodes. All triangulations are shape-/media/arxiv_projects/18759/regular and quasi-uniform; denotes the global mesh size and the number of triangles, which satisfy . For , denote by the space of polynomials of degree at most on an element and by polynomials of degree at most on an edge . Then define

As discrete trial space, we use the conforming lowest-order space

Theorem ? and standard approximation theory combined with the definitions of the norms , by canonical traces, cf. [4], shows that

The optimal test space has finite dimension, but its computation requires to invert the Riesz map in , which has infinite dimension. We will approximate the operator in a finite-dimensional subspace . This approach is called practical DPG method, cf. [23]. We choose to be

where are the edges of the mesh on the boundary. In both examples, we define a domain and prescribe a function . Then, with , we solve with and .

Experiment with smooth solution. In the first example, and is a smooth function. Hence, we expect and observe a convergence rate of .

Experiment with non-smooth solution. In the second example, is an L-shaped domain and with polar coordinates centered at the origin. It follows that for all and so that we expect a convergence rate of . The energy error as well as indeed have order , while we observe an improved order of for .

## 6Conclusion

We presented a numerical method for a transmission problem. The method uses an ultra-weak (finite element) formulation for the interior and standard boundary integral equations for the exterior. The whole system is discretized by a discontinuous Petrov-Galerkin approach and optimal test functions. We obtain quasi-optimality for the field variables and as well as for the trace and normal derivative on the interface . Our analysis builds on the unique solvability of the classical non-symmetric coupling of finite and boundary elements. Numerical experiments support our analysis.

We expect that our method can be extended to other PDEs in the interior for which a DPG analysis is available, e.g., convection-diffusion [18]. Also, based on recent results [19] on unique solvability of the non-symmetric coupling for finite and boundary elements for elasticity problems and DPG finite elements for linear elasticity [6], we expect that our DPG strategy for transmission problems can be extended to problems from linear elasticity.

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