Coupling of the Finite Volume Element Method and the Boundary Element Method

Home , Boundary element method, Finite volume method

COUPLING OF THE FINITE VOLUME ELEMENT METHOD AND THE BOUNDARY ELEMENT METHOD – AN A PRIORI CONVERGENCE RESULT

CHRISTOPH ERATH∗

Abstract. The coupling of the finite volume element method and the boundary element method is an interesting approach to simulate a coupled system of a diffusion convection reaction process in an interior domain and a diffusion process in the corresponding unbounded exterior domain. This discrete system maintains naturally local conservation and a possible weighted upwind scheme guarantees the stability of the discrete system also for convection dominated problems. We show existence and uniqueness of the continuous system with appropriate transmission conditions on the coupling boundary, provide a convergence and an a priori analysis in an energy (semi-) norm, and an existence and an uniqueness result for the discrete system. All results are also valid for the upwind version. Numerical experiments show that our coupling is an efficient method for the numerical treatment of transmission problems, which can be also convection dominated.

Key words. ﬁnite volume element method, boundary element method, coupling, existence and uniqueness, convergence, a priori estimate

1. Introduction. The transport of a concentration in a fluid or the heat propa- gation in an interior domain often follows a diffusion convection reaction process. The influence of such a system to the corresponding unbounded exterior domain can be described by a homogeneous diffusion process. Such a coupled system is usually solved by a numerical scheme and therefore, a method which ensures local conservation and stability with respect to the convection term is preferable. In the literature one can find various discretization schemes to solve similar problems. A very popular method is the coupling of the finite element method (FEM) and the boundary element method (BEM). Here the FEM is applied in the interior domain Ω to solve a problem with inhomogenous material properties and the BEM approximates the solution in the exterior (unbounded) domain Ωe. This avoids the truncation of the unbounded exterior domain, which would be necessary to apply FEM. However, BEM can only be applied to the most important linear partial differential equations with constant coefficients. The first significant results concerning the theoretical justification of a FEM-BEM coupling can be found in [2, 17]. The convergence result of this type was recently extended to Lipschitz coupling interfaces in [22]. A coupling method, which preserves the symmetry of a system can be found in [10]. For an overview of further theoretical developments in context of the FEM- BEM coupling we refer to [16]. But since FEM does not provide local conservation of numerical fluxes in general, we approximate the interior problem by a finite volume element method (FVEM). This method is a well-adapted method for the discretization of various partial differential equations in bounded domains and features local conservation of the numerical fluxes and the natural formulation of an upwind scheme, which ensures stability for the convection part. In a sense BEM is also local conservative. To the best of the author’s knowledge, there is no theoretical justification for any coupling of a finite volume method and the boundary element method available yet. In our system we have a special focus on a convection field in Ω, which divides the coupling boundary naturally in an inflow and outflow part. This is used in the weak representation of the interior problem, where we also replace the interior conormal

∗ National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307, USA ([email protected]) and University of Colorado at Boulder 1 2 CHRISTOPH ERATH

Ωe

Γin b

Ω b

Γout n

Fig. 2.1. Domains and notation for the model problem with the boundary Γ=Γin ∪ Γout. derivative by the exterior conormal derivative. The Calderón system presents the exterior problem in a weak form by an integral equation ansatz, where we replace the exterior trace by the interior trace. We stress that in contrary to the weak formulations in [16, 6] we introduce the exterior trace as an unknown, which will be mandatory for the error analysis of the discrete system in a finite volume sense. A similar approach was used in [3] for an interior Poisson problem approximated by non-conforming finite elements. Then we use the Poincaré Steklov operator in a similar way as in [6] to define another equivalent weak formulation, which allows us to show existence and uniqueness by the Lax-Milgram Theorem. If we replace the interior weak form by a finite volume element discretization and simultaneously approximate the boundary element part by the usual Galerkin method (namely replace the continuous function spaces by discrete ones), we get a 3 × 3 discrete block system of linear equations. For this coupling we prove a convergence result and for sufficient regular solutions an a priori estimate of order O(h), where h is the maximal mesh size. The analysis makes use of a comparison of the standard discrete finite element and the finite volume element bilinear form, similar as in [1, 15, 18, 23], but here with the aid of an energy (semi-) norm related to the above interior problem. After that the Galerkin orthogonality for the boundary element method part and some estimates of the integral operators conclude the proof. We use the convergence result to gain the existence and the uniqueness of a discrete solution. We also define a discrete system with a weighted upwind scheme in the sense of [20] and then use an estimate between the standard and the upwind discrete finite volume element bilinear form to get convergence and an a priori estimate. Existence and uniqueness follow again from this result. The content of this paper is organized as follows. In section 2 we introduce the model problem and some notation. Section 3 gives a short summary on integral equations and the proof of existence and uniqueness of the solution of an equivalent weak form to the model problem. We also define an appropriate energy (semi-) norm. In section 4 we provide the notation for a triangulation and discrete function spaces. We also develop a discrete system, the FVEM-BEM coupling, to approximate the model problem. Section 5 proves a convergence and a priori result and existence and uniqueness of our coupling method. In section 6 we get the same results if we use an upwind method for the convection term in your coupling system. Numerical experiments, found in section 7, conclude the work and confirm the theoretical results with a special focus on convection dominated problems. 2. Model problem and notation. Let Ω ⊂ R2 be a bounded and connected domain with polygonal Lipschitz boundary Γ, see Figure 2.1. In (the interior domain) Coupling FVEM and BEM 3

Ω we consider the following stationary diﬀusion convection reaction problem: Find u such that

(2.1a) div(−A∇u + bu)+ cu = f in Ω,

where A is a symmetric diﬀusion matrix, b is a possibly dominating velocity ﬁeld, c 2 is a reaction function and f is a source term. In the complement Ωe = R \Ω (the exterior domain) we seek ue such that

(2.1b) ∆ue =0 inΩe together with the radiation condition

(2.1c) ue(x)= a∞ + b∞ log |x| + o(1) for |x| → ∞.

We can fix either a∞ ∈ R or b∞ ∈ R and calculate the other one, that means ue behaves asymptotically like the fundamental solution of the Laplace operator, see [19]. Both problems are coupled on the interface Γ = ∂Ω = ∂Ωe, which is closed and has positive surface measure. The coupling boundary Γ is divided in an inflow and outflow part, namely Γin := x ∈ Γ b(x) n(x) < 0 and Γout := x ∈ Γ b(x) n(x) ≥ 0 , respectively, where n is the normal vector on Γ pointing outward with respect to Ω. We allow prescribed jumps u0 and t0 on Γ and demand therefore

(2.1d) u = ue + u0 on Γ, ∂u (2.1e) (A∇u − bu) n = e + t on Γin, ∂n 0 ∂u (2.1f) (A∇u) n = e + t on Γout. ∂n 0 In this work we denote by Lm() and Hm(), m > 0 the standard Lebesgue and Sobolev spaces equipped with the usual norms L2() and Hm(), respectively. For 2 m−1/2 ω ⊂ Ω, (, )ω is the L scalar product. The space H (Γ) is the space of all traces of functions from Hm(Ω) and the duality between Hm(Γ) and H−m(Γ) is given by 2 the extended L scalar product , Γ. The Sobolev space with integral mean zero on m m the boundary is H∗ (Γ) := ψ ∈ H (Γ) ψ, 1Γ . Note that we look for an exterior solution u in H1 (Ω) := v : Ω → R for all K ⊂ Ω compact holds v| ∈ H1(K) . e ℓoc K Furthermore, the Sobolev space W 1,∞ contains exactly the Lipschitz continuous functions. If it is clear from the context, we do not use a notational difference for functions in a domain and its traces. Then the model data are defined as follows: The diffusion matrix A = A(x) is bounded, symmetric and uniformly positive definite, i.e. there exist positive constants 2 T 2 2 CA,1 and CA,2 with CA,1|v| ≤ v A(x)v ≤ CA,2|v| for all v ∈ R and almost every x ∈ Ω. The entries of A(x) are either W 1,∞(Ω) functions or T -piecewise constant functions. Here T denotes a triangulation in triangles of Ω introduced in Subsection 4.1. The convection vector function b ∈ W 1,∞(Ω)2 and the reaction function c ∈ L∞(Ω) satisfy 1 div b(x)+ c(x) ≥ Cb ≥ 0 for almost every x ∈ Ω 2 c,1

with the constant Cbc,1 ≥ 0. We stress that our analysis holds for b constant and c = 0 as well since we allow Cbc,1 = 0. To keep notation clear we assume a∞ =0 in the radiation condition. The model problem reads in a weak sense: 4 CHRISTOPH ERATH

2 1/2 Definition 2.1 (Model problem). For the given data f ∈ L (Ω), u0 ∈ H (Γ) −1/2 1 1 and t0 ∈ H (Γ) we seek u ∈ H (Ω) and ue ∈ Hℓoc(Ωe) satisfying (2.1a)–(2.1f). Remark 2.1. In the interior domain Ω we also could have Dirichlet and Neumann boundary conditions, respectively, where the Neumann boundary is similarly divided as the coupling boundary Γ. 3. Continuous system. In this section we first rewrite the exterior problem in an integral equation. Then we provide a weak form of our model problem and define an energy (semi) norm, which will be used for our further analysis. 3.1. Integral equations. The fundamental solution for the Laplacian (exterior) 1 problem in two dimensional reads − 2π log |x|, which defines formally some linear and bounded integral operators [11, Theorem 1], namely for x ∈ Γ and any s ∈ [−1/2, 1/2] 1 (Vψ)(x)= − ψ(y) log |x − y| ds , V ∈ L Hs−1/2(Γ); Hs+1/2(Γ) , 2π y Γ 1 ∂ (Kθ)(x)= − θ(y) log |x − y| ds , K∈ L Hs+1/2(Γ); Hs+1/2(Γ) , 2π ∂n y Γ y 1 ∂ (K∗ψ)(x)= − ψ(y) log |x − y| ds , K∗ ∈ L Hs−1/2(Γ); Hs−1/2(Γ) , 2π ∂n y Γ x 1 ∂ ∂ (Wθ)(x)= θ(y) log |x − y| ds , W ∈ L Hs+1/2(Γ); Hs−1/2(Γ) . 2π ∂n ∂n y x Γ y Here nx and ny are normal vectors with respect to x and y, respectively. The single layer operator V and the hypersingular integral operator W are symmetric, K is the double layer operator with adjoint K∗. If diam(Ω) < 1, which can always be achieved −1/2 1/2 by scaling, V is H -elliptic and W is H∗ -elliptic, respectively. Therefore, we assume diam(Ω) < 1 in this work. There exists a couple of formulae which characterize the Cauchy data ξ := ue|Γ ∈ 1/2 −1/2 H (Γ) and φ := ∂ue/∂n|Γ ∈ H (Γ) of the exterior problem (2.1b)–(2.1c), e.g. [12]. We only recall two theorems which will be used in the sequel. 1 Theorem 3.1. [12, Lemma 3.5] For ue ∈ Hℓoc(Ωe) with −∆ue = 0 and ue(x)= a∞ + b∞ log |x| + o(1) for |x| → ∞, a∞,b∞ ∈ R there holds

(3.1) ue = −V(φ)+ K(ξ)+ a∞ almost everywhere in Ωe.

Applying the trace and the conormal derivative, respectively, to (3.1) the exterior Calder´on system reads

ξ 1/2+ K −V ξ a (3.2) := + ∞ . φ −W 1/2 −K∗ φ 0

We can ﬁx either a∞ or b∞. For more details on this choice see [19, Theorem 8.9]. Theorem 3.2. [12, Theorem 3.11] The following two statements for (ξ,φ) ∈ H1/2(Γ) × H−1/2(Γ) are equivalent: 1 • ξ,φ are Cauchy data of the problem ue ∈ Hℓoc(Ωe) with −∆ue = 0 in Ωe and ue(x)= a∞ + b∞ log |x| + o(1) for |x| → ∞, a∞,b∞ ∈ R. • There holds (3.2). 3.2. The weak form of the model problem. To gain the weak form of our model problem we test (2.1a) with v, apply integration by parts and replace (A∇u − bu) n and A∇u n by using the conditions (2.1e) and (2.1f), respectively. We also Coupling FVEM and BEM 5

write the Calder´on system (3.2) (a∞ = 0) in a variational way and replace ξ on the left-hand side by the jump condition (2.1d) which leads to: 1 1/2 Definition 3.3 (Weak formulation). Find u ∈ H (Ω), ξ ∈ H∗ (Γ), φ ∈ H−1/2(Γ) such that

(3.3a) A(u,v) −φ,vΓ =(f,v)Ω + t0,vΓ ,

(3.3b) −u,ψΓ − Vφ,ψΓ + (1/2+ K)ξ,ψΓ = −u0,ψΓ , ∗ (3.3c) (1/2+ K )φ,θΓ + Wξ,θΓ = 0

1 1/2 −1/2 for all v ∈ H (Ω), θ ∈ H∗ (Γ), ψ ∈ H (Γ) with

A(u,v):=(A∇u − bu, ∇v)Ω +(cu,v)Ω + b n u,vΓout .

Remark 3.1. From the Calderón system (3.2) we observe with ξ = 1, φ = 0 and a∞ = 1 that W1 = 0 and (1/2+ K)1 = 0. Thus, the variable ξ is determined in (3.3b)–(3.3c) up to an additive constant and we fix this constant by ξ, 1Γ = 0, 1/2 i.e. ξ ∈ H∗ (Γ), see also [4]. Notice that ue is unique because of a∞ = 0 while ξ acts as a layer in the boundary integral operators and is non-unique, but ξ − ue|Γ is constant. The following result can be found similar in [6], which is here extended for the different interior problem and the above weak formulation. To our knowledge this is not found in the literature in this exact form. 1 1 Theorem 3.4. If u ∈ H (Ω), ue ∈ Hℓoc(Ωe) is a solution of the model problem 1 1/2 −1/2 in Definition 2.1, then u ∈ H (Ω), ξ ∈ H∗ (Γ), φ ∈ H (Γ) with ξ := ue|Γ and φ := ∂ue/∂n|Γ solves the weak formulation in Definition 3.3. If, conversely u, ξ, φ is a solution of the weak form in Definition 3.3, then u, ue solves our model problem in 1 Definition 2.1 with ue ∈ Hℓoc(Ωe) defined by

(3.4) ue = −V(φ)+ K(ξ).

Proof. The first direction is clear since the weak form in (3.3a)–(3.3c) is deduced from the model problem in (2.1a)–(2.1f) as described above. Let us prove the other direction. If ue is defined by (3.4) we know from Theorem 3.1 and 3.2, respectively, that ue satisfies (2.1b), (2.1c) and u | 1/2+ K −V u | (3.5) e Γ := e Γ . ∂u /∂n| −W 1/2 −K∗ ∂u /∂n| e Γ e Γ On the other hand, taking the traces of (3.4) we get u | 1/2+ K −V ξ (3.6) e Γ := . ∂u /∂n| −W 1/2 −K∗ φ e Γ The first equation of (3.6) and (3.3b) then show ue|Γ = u − u0 and thus (2.1d). From the second equation of (3.6) and (3.3c) we easily deduce ∂ue/∂n|Γ = φ and thus from (3.5) and (3.6) we get ue|Γ = ξ. Now we apply integration by parts in (3.3a) to get

(div(−A∇u + bu),v)Ω +(cu,v)Ω + (A∇u − bu) n u,vΓ

+ b n u,vΓout −φ,vΓ =(f,v)Ω + t0,vΓ 6 CHRISTOPH ERATH

1 1 1 for all v ∈ H (Ω). If we choose v ∈ H0 (Ω) = v ∈ H (Ω) v|Γ = 0 we get directly the weak form of (2.1a). Hence, if we use (2.1a) in the above equation we get (2.1e) and (2.1f), respectively, because we already showed ∂ue/∂n|Γ = φ. Note that (3.3b) in connection with (2.1d) is equivalent to φ = V−1(−1/2+ K)ξ. If we insert this in (3.3c) we get φ = −Wξ−(1/2−K∗)V−1(1/2−K) =: −Sξ, since V is positive deﬁnite. The (exterior) Poincar´eSteklov operator S : H1/2(Γ) → H−1/2(Γ) is linear, bounded, symmetric and H−1/2-elliptic. By using (2.1d) we replace φ = Su −Su0 in (3.3a) and get so another equivalent weak form of your model problem,

(3.7) 1 B(u,v) := A(u,v)+ Su,vΓ =(f,v)Ω + t0,vΓ + Su0,vΓ for all v ∈ H (Ω), where the right-hand side is a linear bounded functional on H1(Ω). With the Cauchy- Schwarz inequality, the boundedness of S, the trace theorem we can show that the bilinear form B(v,w) is continuous, i.e.

1 B(v,w) ≤ CcontvH1(Ω)wH1(Ω) for all v,w ∈ H (Ω)

with a constant Ccont > 0. The coercivity of B, i.e

2 1 B(v,v) ≥ CcoervH1(Ω) for all v ∈ H (Ω)

for a constant Ccoer > 0, is proven by the assumption of the model parameter, the 1/2 2 2 H -ellipticity of S and the fact that ∇vL2(Ω) + vH1/2(Γ) defines an equivalent norm on H1(Ω). Theorem 3.5. According to Lax-Milgram Theorem there exists a solution for the problems in Definition 2.1 and Definition 3.3 and equation (3.7), and it is unique. 3.3. Energy (semi-) norm. For analytical investigations for diffusion convection reaction problems we define the energy (semi-) norm

1/2 2 1/2 2 1 2 1 (3.8) |||v||| := A ∇v 2 + div b + c v 2 for all v ∈ H (Ω). Ω L (Ω) 2 L (Ω) 1 Then the bilinear form A(v,w) is coercive and for Cbc,1 > 0 continuous on H (Ω) × 1 H (Ω) with respect to ||| |||Ω, i.e.

2 (3.9) A(v,v) ≥ |||v|||Ω,

(3.10) |A(v,w)|≤ CA,2|||v|||Ω|||w|||Ω, and |A(v,w)|≤ CA,2′ |||v|||ΩwH1(Ω).

Here, the constants CA,2,CA,2′ > 0 depend on the data A, b and c and on the in constant Cbc,1. If Cbc,1 = 0 we also require div b + c =0 on Ω and b n =0onΓ that A(v,w) is continuous. 4. Discretization. In this section we develop a FVEM-BEM coupling discretization for the above problem. But ﬁrst we introduce the notation for the triangulation and some discrete function spaces. 4.1. Triangulation. Throughout, T denotes a triangulation of Ω, where N and E are the corresponding set of nodes and edges, respectively. The elements T ∈T are triangles, which are non-degenerated and considered to be closed. For the Euclidean diameter of T ∈T we write hT := diam(T ) := supx,y∈T |x−y|. Moreover, hE denotes Coupling FVEM and BEM 7

V7 V7 V2

τ17 V6 V3 V1

V1 T1 T1 V3 τ13 T1 τ14 T1 V5 τ34

4 V V4

∗ (a) The dual mesh T . (b) Construction of τ17 and DT1 .

Fig. 4.1. Subﬁgure (a) shows the construction of the dual mesh T ∗ from the primal mesh ∗ T . The dashed lines are the new control volumes Vi of T . Subﬁgure (b) shows the interface T1 T1 T1 T1 τ17 = V1 ∩ V7 and the set D = {τ13 ,τ14 ,τ34 }.

∞ the length of an edge E ∈ E. The global mesh size functions hT ,hE ∈ L (Ω) are deﬁned by

hT for x ∈ int(T ), T ∈T , hT : Ω → [0, ∞), x → 0 otherwise, and

hE for x ∈ int(E), E ∈E, hE : E → [0, ∞), x → 0 otherwise, E∈E respectively. The triangulation is regular in the sense of Ciarlet [9], i.e. the ratio of the diameter hT of any element T ∈ T to the diameter of its largest inscribed ball is bounded by a constant independent of hT , the so called shape regularity constant. Additionally, we assume that the triangulation T is aligned with the discontinuities of the coeﬃcients of the diﬀerential equation (if any), the data f, u0 and t0, and the interfaces between Γin and Γout. Throughout, if n appears in a boundary integral, it denotes the unit normal vector to the boundary pointing outward the domain.

Nodes. We introduce the partition N = NΓ ∪NI into all nodes that belong to the coupling boundary NΓ := a ∈N a ∈ Γ and all interior nodes NI := N \NΓ, respectively. For an element T ∈ T we denote by N the set of nodes of T , i.e. T |NT | = 3 for T being a triangle. Additionally, we need for a vertex ai ∈N the index set Ni of all neighbors of ai in N , i.e. all vertices which are connected to ai by an edge E ∈E.

Edges. For the edges we introduce a partition E = EΓ ∪EI into all coupling edges EΓ := E ∈E E ⊂ Γ and all interior edges EI := E\EΓ, respectively. Finally, for an element T ∈ T , we denote by ET ⊂ E the set of all edges of T , i.e. ET := in E ∈E E ⊂ ∂T . The notation in this work is consistent in the sense that EΓ ⊂EΓ denotes all coupling edges on Γin and so on. 8 CHRISTOPH ERATH

Dual mesh. If we connect the center of gravity of an element T ∈ T with the ∗ (edge) midpoint of E ∈ ET we get a dual mesh T see Figure 4.1(a). The dashed lines are the new boxes (elements) V ∈ T ∗ and are considered to be closed and non-degenerated since T consists only elements, which are non-degenerated. A box associated with a vertex ai ∈ N (from the primal mesh T , i = 1 ... #N , which ∗ lies in the box) is denoted by Vi ∈ T . Note that this vertex is unique. The interface between two control volumes Vi and Vj of the dual mesh with Vi ∩ Vj = ∅ T is denoted by τij , i.e. τij = Vi ∩ Vj, whereas τij = Vi ∩ Vj ∩ T is the part of τij , which lies in the corresponding T ∈ T , see Figure 4.1(b). Due to construction τij T or τij can not be a single point. Additionally for all T ∈ T , we define the set T T T ∗ T D := τij τij = Vi ∩ Vj ∩ T for Vi,Vj ∈T with Vi = Vj . Note that |D | = 3 for all T ∈T . Discrete function spaces. We define the piecewise affine and global continuous 1 function space on T by S (T ) := v ∈C(Ω) v|T affine for all T ∈T . On the dual mesh T ∗ we provide P0(T ∗) := v ∈ L2(Ω) v| constant V ∈T ∗ . With the aid of V the characteristic function χ∗ over the volume V we write for v∗ ∈P0(T ∗) i i ∗ ∗ ∗ v := vi χi . xi∈N 1 0 The spaces S (EΓ) and P (EΓ) are equivalently defined as above related to Γ and 1 S∗ (EΓ) is the piecewise affine global continuous function space with integral mean ∗ zero over EΓ. Furthermore, we define the T -piecewise interpolation operator

∗ 0 ∗ ∗ ∗ (4.1) Ih : C(Ω) →P (T ), Ihv := v(ai)χi (x). ai∈N 1 Lemma 4.1. Let T ∈T and E ∈ET . For vh ∈S (T ) there holds

∗ (4.2) (vh −Ihvh) dx = 0, T ∗ (4.3) (vh −Ihvh) ds = 0, E ∗ (4.4) vh −IhvhL2(T ) ≤ C1hT ∇vhL2(T ), 1/2 ∗ 2 2 (4.5) vh −IhvhL (E) ≤ C2hE ∇vhL (T ),

where the constants C1,C2 > 0 depend only on the shape regularity constant. ∗ Proof. The proof is based on the construction of T and is simple, since vh is a piecewise linear function on T , see [8, 23]. Additionally, we deﬁne the ‘broken Sobolev space’ on EΓ, i.e.

m 2 m H (EΓ) := v ∈ L (Γ) v|E ∈ H (E) for all E ∈EΓ for m> 0. 2 4.2. The discrete system. For technical reasons we assume t0 ∈ L (Γ). In general a ﬁnite volume scheme integrates the model equation over control volumes and transforms this surface integral partly into its boundary. If we integrate (2.1a) over each dual element V ∈T ∗, apply the divergence theorem, (2.1e) and (2.1f) (with Coupling FVEM and BEM 9

ξ := ue|Γ and φ := ∂ue/∂n|Γ) we get a balance equation for the interior problem

(−A∇u + bu) n ds + cudx (4.6) ∂V \Γ V

+ b n uds − φds = f dx + t0 ds out ∂V ∩Γ ∂V ∩Γ V ∂V ∩Γ for all V ∈ T ∗. We can write the ﬁnite volume element part of the left-hand side of (4.6) as bilinear form over all w ∈ H1(Ω) and all v∗ ∈P0(T ∗):

∗ ∗ AV (w,v ) := vi (−A∇w + bw) n ds + cw dx ∂Vi\Γ Vi (4.7) ai∈N + b n wds . out ∂Vi∩Γ Then the right-hand side reads

∗ ∗ (4.8) F (v ) := vi f dx + t0 ds . Vi ∂Vi∩Γ ai∈N To discretize (4.6), we approximate the solution u ∈ H1(Ω) of (2.1a) in the interior 1 domain by uh ∈ S (T ) in a conforming ﬁnite element space. The approximation of the exterior part, namely (3.3b)–(3.3c), is based on the replacement of the continuous spaces by suitable discrete spaces and so we get the usual (discrete) variational 1 1/2 0 formulation, i.e. S∗ (EΓ) is the piecewise aﬃne approximation of H∗ (Γ) and P (EΓ) is the piecewise constant approximation of H−1/2(Γ), respectively. Definition 4.2 (Discrete problem). Additionally to the previous data assump- 2 1 1 0 tion, we demand t0 ∈ L (Γ). Find uh ∈ S (T ), ξh ∈ S∗ (EΓ) and φh ∈ P (EΓ) such that

∗ ∗ ∗ (4.9a) AV (uh,v ) − (φh,v )Γ = F (v )

(4.9b) −uh,ψhΓ − Vφh,ψhΓ + (1/2+ K)ξh,ψhΓ = −u0,ψhΓ , ∗ (4.9c) (1/2+ K )φh,θhΓ + Wξh,θhΓ = 0

∗ 0 ∗ 1 0 for all v ∈P (T ), θh ∈S∗ (EΓ), ψh ∈P (EΓ). We prove existence and uniqueness of the discrete system (4.9a)–(4.9c) with the aid of a convergence result in Corollary 5.4. 5. An a priori convergence result. For technical reasons we assume φ ∈ 2 2 L (Γ) slightly more regular, see Remark 5.1, and t0 ∈ L (Γ). We deﬁne the error 1 1/2 e := u − uh ∈ H (Ω) in the interior domain, the trace error δ := ξ − ξh ∈ H (Γ) 2 and the conormal error ǫ := φ − φh ∈ L (Γ) of the exterior problem. From (3.3b)–(3.3c) and (4.9b)–(4.9c) we easily verify a Galerkin orthogonality of the exterior part and deﬁne therefore

0 (5.1) p0 := −e −Vǫ + (1/2+ K)δ ⊥P (EΓ), ∗ 1 (5.2) p1 := (1/2+ K )ǫ + Wδ ⊥S∗ (EΓ).

In the following we get an estimate for the right-hand side f. 10 CHRISTOPH ERATH

Lemma 5.1. There holds for the right-hand side f

∗ 2 2 (5.3) | (f,vh −Ihvh)Ω |≤ C hT fL (T )∇vhL (T ) T∈T 1 for all vh ∈S (T ) with a constant C > 0, which depends only on the shape regularity constant. Proof. The proof is simple and can be found in [23]. The next lemma gives us an estimate between the weak and the ﬁnite volume 1 element bilinear form for a function vh ∈S (T ) in the energy (semi-) norm. in Lemma 5.2. Let us assume that bn is piecewise constant on Γ , i.e. bn|Γin ∈ 0 in 1 P (EΓ ). If Cbc,1 = 0 we also require div b + c = 0 on Ω. For all vh,wh ∈ S (T ) there holds

∗ (5.4) |A(vh,wh) −AV (vh, Ihwh)|≤ C hT |||vh|||T ∇whL2(T ) T∈T with a constant C > 0, which depends only on the model data A, b, c and the shape regularity constant. Proof. As in [23] we rewrite the bilinear form A(vh,wh) by applying integration by parts to

A(vh,wh)= − (div(A∇vh − bvh),wh)T +(A∇vh n,wh)∂T T∈T +(cvh,wh)T +(b n vh,wh)∂T ∩Γin .

∗ ∗ Let us deﬁne wh := Ihwh. We rewrite the ﬁnite volume element bilinear form ∗ AV (vh,wh) in a similar way (5.5)

∗ ∗ AV (vh,wh)= (−A∇vh + bvh) n wh ds T ∩(∂Vi\Γ) T∈T ai∈NT ∗ ∗ + cvhwh ds + b n vhwh ds . T ∩V T ∩(∂V ∩Γout) i i Note that with integration by parts we can write

∗ (−A∇vh + bvh) n wh ds T ∩(∂Vi\Γ) ai∈NT ∗ ∗ = (−A∇vh + bvh) n wh ds − (−A∇vh + bvh) n wh ds ∂(T ∩Vi) ∂T ai∈NT ∗ ∗ (5.6) = div(−A∇vh + bvh)wh dx − (−A∇vh + bvh) n wh ds. T ∂T ∗ Thus, (5.5) and the fact that wh does not jump across the edges E leads to

∗ ∗ ∗ AV (vh,wh)= − (div(A∇vh − bvh),wh)T +(A∇vh n,wh)∂T (5.7) T∈T ∗ ∗ +(cvh,wh)T − (b n vh,wh)∂T ∩Γin . Coupling FVEM and BEM 11

Note that if A is T -piecewise constant, all parts with A vanish in A(vh,wh) − ∗ AV (vh,wh) because of div(A∇vh) = 0 and (4.3). This is well-known, see e.g. [1, 15]. 1,∞ Thus the following shows the estimation for A with entries in W (Ω). Since ∇vh is 0 in constant and div(bvh) = div(b)vh+b∇vh on T and by assumption bn|Γin ∈P (EΓ ) we get with (4.3) ∗ A(vh,wh) −AV (vh,wh) ∗ = (− div(A)∇vh + div(b)vh + b ∇vh)+ cvh,wh − wh)T T∈T ∗ + (A − A)∇vh n,wh − wh E E∈E T ∗ − (b n(vh − vh),wh − wh)E . in E∈ET ∩Γ 2×2 Here, div(A) := (div(a11,a21), div(a12,a22)) with the entries aij of A, A ∈ R denotes a constant matrix, where the entries are the integral means of aij over E, 0 and vh ∈ P (EΓ) is EΓ-piecewise constant approximation of vh. Next we apply the ′ 1,∞ Poincar´einequality g − gL∞(E) ≤ CP hEg L∞(E) (CP > 0, g ∈ W (E) and g the integral mean of g over E) for the entries of A − A. If we additionally use the Cauchy-Schwarz inequality we deduce ∗ |A(vh,wh) −AV (vh,wh)| ∗ ∗ ≤ CT |||vh|||T wh − whL2(T ) + hE∇vhL2(E)wh − whL2(E) T∈T E∈ET ∗ + vh − vhL2(E)wh − whL2(E) in E∈ET ∩EΓ

with the constant CT > 0, which depends on the shape regularity constant, CA,1 and Cbc,1, if Cbc,1 > 0. Note that this estimate is valid for Cbc,1 = 0 as well because then −1/2 2 2 div b + c = 0. A simply calculation proves ∇vhL (E) ≤ C1hE ∇vhL (T ) with 1/2 2 2 C1 > 0. Together with the trace inequality vh − vhL (E) ≤ C2hT ∇vhL (T ) with C2 > 2 and (4.4)–(4.5) we conclude the proof. This leads us to the main result in this section, a convergence and a priori result for the discrete solution. Theorem 5.3 (A priori convergence estimate). Let b n be piecewise constant in 0 in on Γ , e.g. b n|Γin ∈ P (EΓ ). For Cbc,1 = 0 we also require div b + c = 0 on in 1 1/2 2 Ω and b n = 0 on Γ . For the solution u ∈ H (Ω), ξ ∈ H∗ (Γ), φ ∈ L (Γ) of 1 our model problem in Deﬁnition 2.1 there holds with a discrete solution uh ∈S (T ), 1 0 ξh ∈ S∗ (EΓ), φh ∈ P (EΓ) of the discrete problem in Deﬁnition 4.2 and hT small enough

|||u − uh|||Ω + ξ − ξhH1/2(Γ) + φ − φhH−1/2(Γ) 1/2 1/2 2 2 2 ≤ C hT fL (Ω) + |||hT u|||Ω + hE (t0 − t0)L (Γ) + hE (φ − φ)L (Γ) + inf u − vhH1(Ω) + inf ξ − ξhH1/2(Γ) , 1 e 1 vh∈S (T ) ξh∈S∗ (EΓ) where φ and t0 is the EΓ-piecewise integral mean of φ and t0, respectively, and C > 0 is a constant, which depends on the model data A, b and c and the shape regularity 12 CHRISTOPH ERATH

constant. 2 1 2 2 1/2 Furthermore, for u ∈ H (Ω), ξ ∈ H∗ (Γ) ∩ H (EΓ), φ ∈ L (Γ) ∩ H (EΓ) and t0 ∈ 2 1/2 L (Γ) ∩ H (EΓ) we get

|||u − uh|||Ω + ξ − ξhH1/2(Γ) + φ − φhH−1/2(Γ) = O(h)

with h := maxT ∈T {hT }. 1 Proof. From (3.9) we estimate with an arbitrary vh ∈S (T ) and e = u − uh

2 |||e|||Ω ≤A(e,u − vh)+ A(e,vh − uh).

The ﬁrst term of the right-hand side can be easily estimated by (3.10)

′ 1 2 −1 2 |A(e,u − vh)|≤ CA,2 |||e|||Ωu − vhH (Ω) ≤ C1|||e|||Ω + C1 u − vhH1(Ω), where we can ﬁx C1 > 0 because of Young’s inequality. For the A(u,vh − uh) we use ∗ the identity (3.3a) and insert (4.9a) with Ih(wh − uh) to obtain

A(u,vh − uh) −A(uh,vh − uh) ∗ ∗ =(f,vh − uh −Ih(vh − uh))Ω −A(uh,vh − uh)+ AV (uh, Ih(vh − uh)) (5.8) ∗ +(t0,vh − uh −Ih(vh − uh))Γ + φ − φh,vh − uhΓ ∗ +(φh,vh − uh −Ih(vh − uh))Γ .

Now we estimate these terms separately. With Lemma 5.1 and 5.2 we calculate

∗ ∗ (f,vh − uh −Ih(vh − uh))Ω −A(uh,vh − uh)+ AV (uh, Ih(vh − uh))

≤ C2 hT fL2(T ) + hT |||u|||T + hT |||u − uh|||T ∇(vh − uh)L2(T ) T∈T −1 2 2 2 ≤ C3 hT fL2(Ω) + |||hT u|||Ω + |||hT (u − uh)|||Ω 2 2 + C3∇(u − vh)L2(Ω) + C3|||u − uh|||Ω.

We have used Young’s inequality in the last step, where we can ﬁx the constant 2 C3 > 0. Note that t0 ∈ L (Γ) thus we get with (4.3), Cauchy-Schwarz inequality and (4.5)

1/2 ∗ 2 2 | (t0,vh − uh −Ih(vh − uh))Γ |≤ t0 − t0L (E)hE ∇(vh − uh)L (TE ). E∈EΓ

Here, TE is the element associated with E. The same calculations as above lead to

∗ | (t0,vh − uh −Ih(vh − uh))Γ | −1 1/2 2 2 2 ≤ C4 hE (t0 − t0)L2(E) + C4∇(u − vh)L2(Ω) + C4|||u − uh|||Ω,

where we can ﬁx C4 > 0 arbitrary. The H¨older inequality and the trace theorem, i.e. u − vhH1/2(Γ) ≤ C5u − vhH1(Ω) with C5 > 0, lead to

φ − φh,vh − uhΓ = φ − φh,vh − uΓ + φ − φh,u − uhΓ

1 ≤ C5φ − φhH−1/2(Γ)u − vhH (Ω) + φ − φh,u − uhΓ . Coupling FVEM and BEM 13

∗ From the deﬁnition of p0 in (5.1) and p1 in (5.2), and because K is adjoint to K we write with δ = ξ − ξh and ǫ = φ − φh

(5.9)

(φ − φh,u − uh)Γ =(ǫ, (1/2+ K)δ)Γ − (ǫ, Vǫ)Γ − (ǫ,p0)Γ 2 2 ≤−CV ǫH−1/2(Γ) − CW δH1/2(Γ) −p0,φ − φΓ + p1,ξ − ξhΓ,

1 where φ is the EΓ-piecewise integral mean of φ and ξh ∈ S∗ (EΓ) is chosen arbitrary. Note that CV > 0 and CW > 0 are the ellipticity constants from the operators V and W, respectively. Next we estimate

(5.10)

1 2 −1 2 − p0,φ − φ Γ = e − eE,φ − φ Γ + Vǫ − (1/2+ K)δH / (Γ)φ − φH / (Γ), where eE is the EΓ-piecewise integral mean of e on Γ. We calculate with the trace 1/2 2 ′ 2 ′ inequality e − eEL (Γ) ≤ C6 hE ∇eL (Ω) with C6 > 0 and continuity of V and K

1/2 2 2 − p0,φ − φ Γ ≤ C6 ∇eL (Ω)hE (φ − φ)L (Γ) + ǫH−1/2(Γ) + δH1/2(Γ) φ − φH−1/2(Γ) 2 −1 1/2 2 ≤ C7|||e|||Ω + C7 hE (φ − φ)L2(Γ) 2 2 −1 1/2 2 + C8ǫH−1/2(Γ) + C8δH1/2(Γ) + C8 hE (φ − φ)L2(Γ),

where we have used Young’s inequality in the last estimation and [7, Lemma 4.3], 1/2 ′ 2 ′ i.e. φ − φH−1/2(Γ) ≤ C8 hE (φ − φ)L (Γ) with C8 > 0, for the last term. Note that we can ﬁx C7,C8 > 0. Similarly as above we estimate

∗ −1 2 1 2 −1 2 p1,ξ − ξh Γ ≤ (1/2+ K )ǫH / (Γ) + WδH / (Γ) ξ − ξhH / (Γ) 2 2 −1 2 ≤ C9ǫ −1/2 + C9δ 1/2 + C ξ− ξh 1/2 , H (Γ) H (Γ) 9 H (Γ) ∗ which follows from the continuity of K and W, C9 > 0. Now we ﬁx the constants C8 and C9 such that C10 := CV − C8 − C9 > 0 and C11 := CW − C8 − C9 > 0. Thus, we get for (5.9)

2 2 2 (φ − φh,u − uh)Γ ≤−C10ǫH−1/2(Γ) − C11δH1/2(Γ) + C7|||e|||Ω −1 −1 1/2 2 −1 2 + max C7 ,C8 hE (φ − φ)L2(Γ) + C9 ξ − ξhH1/2(Γ). For the last term in (5.8) we may apply (4.3)

∗ (φh,vh − uh −Ih(vh − uh))Γ = 0

0 since φh ∈P (EΓ). If we choose C1,C3,C4,C7 < 1 we conclude the proof for hT small −1 2 enough (because of C3 |||hT (u − uh)|||Ω). The second assertion follows directly from the ﬁrst by applying the approximation theorem. Remark 5.1. Note that we need the additional regularity φ ∈ L2(Γ) to estimate b −1/2 e,φ − φ Γ in (5.10). If we only allow C c,1 > 0, then φ ∈ H (Γ) is suﬃcient to prove Theorem 5.3. This can be seen by the following estimations. Let us choose 14 CHRISTOPH ERATH

0 φh ∈P (EΓ) arbitrary. Then we get e,φ − φh Γ instead of e,φ − φ Γ, which follows easily by (5.1). We estimate ′ 1 2 −1 2 1 −1 2 e,φ − φh Γ ≤ eH / (Γ)φ − φhH / (Γ) ≤ C eH (Ω)φ − φhH / (Γ) ′′ ≤ C |||e|||Ωφ − φh −1/2 , H (Γ) ′ ′ where we have used the trace theorem, i.e. vH1/2(Γ) ≤ C vH1(Ω) with C > 0, and the fact that in this case we can estimate the H1-norm by the energy norm with C′′ > 0. Thus we are ﬁnished if we replace

1/2 h (φ − φ)L2(Γ) by inf φ − φhH−1/2(Γ) E e 0 φh∈P (EΓ) in the rest of the proof. With the aid of Theorem 5.3 we prove the existence and uniqueness of the discrete problem in Definition 4.2. in 0 in Corollary 5.4. Let b n be piecewise constant on Γ , e.g. b n|Γin ∈P (EΓ ). in For Cbc,1 = 0 we also require div b + c = 0 on Ω and b n = 0 on Γ . The 1 1 0 discrete solution uh ∈ S (T ), ξh ∈ S∗ (EΓ), φh ∈ P (EΓ) of the discrete problem in 1 1/2 −1/2 Definition 4.2 to the solution u ∈ H (Ω), ξ ∈ H∗ (Γ), φ ∈ H (Γ) of our model problem in Definition 2.1 exists and is unique. Proof. It is easy to see that the discrete problem in Definition 4.2 leads to a linear system of Ns = #N + #EΓ + #NΓ equations with Ns unknowns. Let us assume that uh, ξh and φh is a solution of the system with the right-hand side 0. We can write this Ns×Ns system as As xs = 0 with the system matrix As ∈ R and the unknown vector Ns×1 xs ∈ R , which consists the unknowns of uh, ξh and φh. For the data f = 0, u0 = 0 and t0 = 0 we observe by the equivalent weak formulation in Definition 3.3 that u = 0, ξ = 0 and φ = 0 hold and this continuous solution is unique because of Theorem 3.5. Obviously uh, ξh and φh is a discrete solution of this continuous system and Theorem 5.3 holds. Therefore, we get with the chosen data

(5.11) |||uh|||Ω + ξhH1/2(Γ) + φhH−1/2(Γ) ≤ 0.

1/2 2 Note that we can estimate |||uh|||Ω ≥ CA,1∇uhL (Ω). Thus, we get from (5.11) that ξh = 0, φh = 0 and ∇uh = 0, which implies that uh is constant in Ω. From (4.9b) we 0 observe with ξh = 0 and φh = 0 that uh,ψhΓ = 0 for all ψh ∈P (EΓ), in particular the integral mean of uh on the boundary Γ is zero and thus uh = 0 on Ω. That means As is injective and thus bijective (since As is square), which proves the existence and uniqueness of the discrete solution. Remark 5.2. Note that for Corollary 5.4 we do not need additionally regularity for φ, neither for Cbc,1 = 0. 6. The Discrete Problem with an Upwind Approximation. For singularly perturbed diffusion convection problems, i.e. the diffusion is small with respect to the convection vector, the above approximation leads to oscillating numerical results. This disappointing behavior occurs because such methods lose stability and cannot adequately approximate solutions inside layers. In the context of finite element methods the streamline diffusion finite element method (SDFEM) is often used. For finite volume schemes an upwind method is natural, which preserves the local ∗ conservation and gives the desired stabilization. Therefore, we replace AV (uh,v ) Coupling FVEM and BEM 15

in (4.9a) (the ﬁnite volume element part) by

up ∗ ∗ AV (uh,v ) := vi −A∇uh n ds + cudx ∂Vi\Γ Vi ai∈N (6.1) T + b niuh,ij ds + b n uh ds . T out τ ∂Vi∩Γ j∈Ni τ T ⊂τ ij ijij We refer to Figure 4.1(b) for the notation and remark that there are exactly two T T τij ⊂ τij and ni is the outer unit normal vector to ∂Vi. The discrete function uh,ij T T T is defined as uh,ij := λij uh(ai)+(1 − λij)uh(aj ). There are many possibilities how T T T T T to define the weights λij. We consider λij := Φ(βij |τij |/Aij ) with the weighting function Φ : R → [0, 1], which gets the Péclet number as argument, see e.g. [20]. Here, we calculate

T 1 βij := T b ni ds |τ | τ T ij ij T and assume that b ni does not change sign over τij . The row sum norm of the T integral means akl, k,l = 1, 2 of the entries of A over τij is

T Aij := max |a11| + |a12|, |a21| + |a22| . In this work we choose the classical (full) upwind scheme by

(6.2) Φ(t) = (sign(t)+1)/2,

T T T i.e. λij = 1 for βij ≥ 0 and λij = 0 otherwise. The second choice is

min 2|t|−1, 1 /2 for t< 0, (6.3) Φ(t) := −1 1 − min 2|t| , 1 /2 for t ≥ 0. where we can steer the amount of upwinding to reduce the excessive numerical diﬀu- sion, which is also successfully used in [13]. The proof for existence and uniqueness of the (upwind) coupling system (4.9a)– up ∗ ∗ (4.9c), with AV (uh,v ) instead of AV (uh,v ), can be found in Corollary 6.3. First we will prove the following relationship. 1 Lemma 6.1. For all vh,wh ∈S (T ) there holds

up ∗ ∗ 2 |AV (vh, Ihwh) −AV (vh, Ihwh)|≤ C hT |||vh|||T ∇whL (T ) T∈T with a constant C > 0, which depends on the model data A, b, c and the shape regularity constant. ∗ ∗ Proof. Let us deﬁne wh := Ihwh. With the bilinear forms (4.7) and (6.1) we get

∗ up ∗ AV (vh,wh) −AV (vh,wh)

∗ T = wi b ni vh ds − b ni vh,ij ds . T τij τ ai∈N j∈Ni τ T ⊂τ ij ijij 16 CHRISTOPH ERATH

We can express this sum over the elements of T .

∗ up ∗ AV (vh,wh) −AV (vh,wh)

∗ ∗ T = (wi − wj ) b ni(vh − vh,ij ) ds. τ T T ∈T τ T ∈DT ij ij

T T T T Note that vh,ij = λij vh(ai)+(1 − λij )vh(aj ) with λij ∈ [0, 1]. With the Cauchy- Schwarz inequality we get

∗ ∗ T ∗ ∗ T (w − w ) b ni(vh − v ) ds ≤ b ni ∞ T w − w 2 T vh − v 2 T i j h,ij L (τij ) i j L (τij ) h,ij L (τij ) τ T ij ≤ ChT ∇wh 2 T hT ∇vh 2 T . L (τij ) L (τij )

T Note that vh and wh are linear on τij which leads to the last estimation. The constant T C > 0 depends on the weighting factor λij and b. We easily conclude the proof by the −1/2 −1/2 estimates ∇wh 2 T ≤ h ∇wh 2 and ∇vh 2 T ≤ h ∇vh 2 . L (τij ) T L (T ) L (τij ) T L (T ) Similar as in Theorem 5.3 we state an a priori result for the coupling with upwinding. Theorem 6.2 (A priori convergence estimate for upwinding). Let us assume in 0 int that b n is piecewise constant on Γ , e.g. b n|Γin ∈P (EΓ ). If Cbc,1 = 0 we also require div b + c = 0 on Ω and b n = 0 on Γin. For the solution u ∈ H1(Ω), ξ ∈ 1/2 2 H∗ (Γ), φ ∈ L (Γ) of our model problem in Deﬁnition 2.1 there holds with a discrete 1 1 0 solution uh ∈ S (T ), ξh ∈ S∗ (EΓ), φh ∈ P (EΓ) of the discrete (upwind) problem in up ∗ ∗ Deﬁnition 4.2 (with AV (uh,v ) instead of AV (uh,v )) and hT small enough

|||u − uh|||Ω + ξ − ξhH1/2(Γ) + φ − φhH−1/2(Γ) 1/2 1/2 2 2 2 ≤ C hT fL (Ω) + |||hT u|||Ω + hE (t0 − t0)L (Γ) + hE (φ − φ)L (Γ) + inf u − vhH1(Ω) + inf ξ − ξhH1/2(Γ) , 1 e 1 vh∈S (T ) ξh∈S∗ (EΓ) where φ and t0 is the EΓ-piecewise integral mean of φ and t0, respectively, and C > 0 is a constant, which depends on the model data and the shape regularity constant. 2 1 2 2 1/2 Furthermore, for u ∈ H (Ω), ξ ∈ H∗ (Γ) ∩ H (EΓ), φ ∈ L (Γ) ∩ H (EΓ) and t0 ∈ 2 1/2 L (Γ) ∩ H (EΓ) we get

|||u − uh|||Ω + ξ − ξhH1/2(Γ) + φ − φhH−1/2(Γ) = O(h)

with h := maxT ∈T {hT }. Proof. The proof follows exactly the lines of the proof of Theorem 5.3. With 1 up ∗ vh ∈ S (T ) we get in the same way (5.8), now with AV (uh, Ih(vh − uh)) instead ∗ ∗ ∗ of AV (uh, Ih(vh − uh). Thus, we plug in AV (uh, Ih(vh − uh)) − AV (uh, Ih(vh − ∗ uh)) in (5.8). Note that we can estimate −A(uh,vh − uh)+ AV (uh, Ih(vh − uh)) by up ∗ ∗ Lemma 5.2 and AV (uh, Ih(vh − uh)) −AV (uh, Ih(vh − uh)) with Lemma 6.1 and the other terms as in the proof of Theorem 5.3. Remark 6.1. We refer to Remark 5.1 for a discussion on φ ∈ H−1/2(Γ). With the aid of Theorem 6.2 we prove the existence and uniqueness of the discrete problem in Deﬁnition 4.2 with upwinding. Coupling FVEM and BEM 17

in 0 in Corollary 6.3. Let b n be piecewise constant on Γ , e.g. b n|Γin ∈P (EΓ ). in For Cbc,1 = 0 we also require div b + c = 0 on Ω and b n = 0 on Γ . The discrete 1 1 0 solution uh ∈ S (T ), ξh ∈ S∗ (EΓ), φh ∈ P (EΓ) of the discrete (upwind) problem in up ∗ ∗ 1 Deﬁnition 4.2 (with AV (uh,v ) instead of AV (uh,v )) to the solution u ∈ H (Ω), 1/2 −1/2 ξ ∈ H∗ (Γ), φ ∈ H (Γ) of our model problem in Deﬁnition 2.1 exists and is unique. Proof. The proof is the same as for Corollary 5.4. 7. Numerical results. In this section we provide two numerical examples to test our coupling method, especially with an upwind scheme. The implementation is straightforward, we only want to remark a few points. We do not need to store

the dual mesh for FVEM. The calculation of the Galerkin matrix Vφh,ψhΓ follows the result of [5] by use of analytic anti-derivatives. There, an analytical expression for the entries of the Galerkin matrix is provided. The same technique is available for the Galerkin matrix Kξh,ψhΓ. We stress that both matrices are dense and thus the assembly is of quadratic complexity. For the other two BEM matrices we use 1/2 ∗ Wψ,θΓ = V(∂ψ/∂s),∂θ/∂sΓ for ψ,θ ∈ H (Γ), see [19], and the fact that K is the adjoint of K. If we know the analytical solution, we calculate the energy norm by

1/2 2 2 2 Eh := |||u − uh|||Ω + |||φ − φh|||V + |||ξ − ξh|||W . Note that Eh represent the error of Theorem 5.3 and Theorem 6.2, respectively, because of the equivalence of the norms, i.e.

2 2 ξ − ξhH1/2(Γ) ∼ |||ξ − ξh|||W := W(ξ − ξh),ξ − ξhΓ , 2 2 φ − φhH−1/2(Γ) ∼ |||φ − φh|||V := V(φ − φh),φ − φhΓ .

We follow again the ideas of [5] to calculate |||φ−φh|||V , which leads to an approximation of a double integral and where we differ two cases in order to calculate the outer integral. We do not want to go into details. The energy norm |||ξ − ξh|||W is again calculated by the relation between the single layer and hypersingular integral operator. In all experiments, the initial mesh consists of triangles of about the same size. 7.1. Diffusion Convection Problem. We consider the model problem on the square domain Ω = (0, 1/2) × (0, 1/2). We choose a fixed diffusion matrix of A ∈ T {0.05 I, 1000 I}, a convection field b = (100x1, 0) and a reaction coefficient c = 0. Note that for this problem the coupling condition (2.1e) does not occur, i.e. we have no inflow boundary Γin. For all calculations we use the upwind discrete coupling with the weighting function Φ defined in (6.3). We prescribe an analytical solution

0.25 − x u(x ,x ) = 0.5 1 − tanh 1 1 2 0.02 for the interior domain Ω and

2 2 ue(x1,x2) = log (x1 − 0.25) +(x2 − 0.25) for the exterior domain Ωe with a∞ = 0 and b∞ = 1. We calculate the right-hand side f and the jumps u0 and t0 appropriate. In Figure 7.1 we plot the convergence rate for uniform mesh-reﬁnement for A = 0.05 I (lower part) and A = 1000 I (upper 18 CHRISTOPH ERATH

2 10

1 10 1 1/2

0 10 error

−1 10 Eh (A = 1000 I)

1/2 Eh (A =0.05 I)

−2 1 10

1 2 3 4 5 6 10 10 10 10 10 10 number of elements

1 2 2 2 2 / Fig. 7.1. Energy error Eh = |||u − uh|||Ω + |||φ − φh|||V + |||ξ − ξh|||W in the example in Subsection 7.1 for uniform mesh-reﬁnement for A = {0.05 I, 1000 I}.

1 0.5 0.5

−0.5 0 −1

−1.5 0.75 −0.5

0.5

0.25 −1 0.75 0.5 0 0.25 x2 0 −0.25 −0.25 x1

Fig. 7.2. Solution on a uniformly generated mesh with 4096 elements in the example in Sub- section 7.1 for A = 0.05 I. part) with respect to the number of elements. Here, both axes are scaled logarith- mically. Therefore, a straight line g with slope −p corresponds to a dependence Coupling FVEM and BEM 19

0.25 0.3

0.15 2 2 0 x x −0.05 −0.15 f =5

−0.2 −0.3

−0.2 −0.1 0 0.25 −0.3 −0.15 0 0.15 0.3 x1 x1 (a) Volume force f. (b) Contour lines.

Fig. 7.3. Domain Ω=(−1/4, 1/4)2\ [0, 1/4] × [−1/4, 0] and volume force f in (a) in the example in Subsection 7.2. The volume force f has the value 5 in the gray rectangle, otherwise it is 0. We generate the contour lines in (b) from the solution of the uniformly generated mesh with 196008 elements.

0.06 0.03 0.05 0.05 0.05 0.04 0.025 0.04 0.04 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.01 0.015 0.01 0.01 0 0 0 0.01 −0.01 −0.01 0.25 −0.01 0.25 −0.02 0.005 −0.02 0 −0.02 0 −0.25 −0.25 −0.03 −0.25 −0.25 0 0.25 x2 0 0.25 x2 x1 x1 (a) Solution with oscillations. (b) Solution with full upwinding.

Fig. 7.4. Strong oscillations of the interior solution by approximation without upwinding in (a) compared with the full upwind scheme solution in (b) in the example in Subsection 7.2 for uniform mesh-reﬁnement with 3072 elements. g = O(N −p), where N = #T denotes the number of elements. Note that for uniform mesh-reﬁnement, the order O(N −p) with respect to N corresponds to O(h2p) with respect to the maximal mesh size h := max hT . Since the interior and exterior solution T ∈T are smooth, we observe the expected convergence rate of O(N −1/2) in both cases, see Theorem 6.2. In Figure 7.2 we plot the interior and exterior solution. For the exterior solution we evaluate the exterior representation formula (3.1) on a uniform grid with 1536 triangles on each node with the Cauchy data from an interior mesh with 4096 elements. For points on the boundary Γ coming from the exterior domain, we use the exterior trace of (3.1). Note that this trace reads ϕ ξ (x)= −(Vφ )(x)+ K + ξ (x)+ a h h 2π h ∞ for a point evaluation x ∈ Γ, where ϕ is the interior angle of the intersection of the two tangential vectors in x. According to Remark 3.1 the left-hand side gives us the approximative value of ξ. 20 CHRISTOPH ERATH

7.2. Convection dominated problem. For our model problem we consider the classical L-shaped domain Ω = (−1/4, 1/4)2\ [0, 1/4] × [−1/4, 0] as shown in Figure 7.3(a). The diﬀusion matrix A = α I in Ω is piecewise constant and reads −4 10 for x2 ≤ 0, R R R −3 α : × → :(x1,x2) → 10 for x1 > 0,  5 10−4 else.  Additionally, we choose b = (15, 10)T and c = 10−2. We have a volume force f in the lower square, i.e.

5 for − 0.2 ≤ x1 ≤−0.1, −0.2 ≤ x2 ≤−0.05, f = 0 else,

see also Figure 7.3(a). We prescribe the jumps u0 = 0 and t0 = 0 and and the radiation condition b∞ = 0. To fix b∞ in general, the integral mean of φ over Γ has to be 2πb∞, see [14, Lemma 2.1] for more details. We use the full upwind scheme for the approximation of the convection term. This model can describe the stationary concentration of a chemical dissolved and distributed in different fluids, where we have a convection dominated problem in the interior Ω and a diffusion distri- bution in the exterior domain Ωe. The solution of such a problem may have local phenomena such as injection wells and will lead to step layers on the boundary (0, 0) to (0, −0.25), due to the convection in this direction and the different diffusion coefficient of the interior and exterior problem. Note that this problem is convection dominated and we only could get a solution without oscillation by use of the full upwind scheme, i.e. the weighting function (6.2), instead of partly upwinding by the weighting function (6.3). The reason might be the steep layer on the boundary from (0, 0) to (0, −0.25). Figure 7.4(a) shows the interior approximation without an upwind scheme for the convection part on a uniform generated mesh with 3072 elements, which leads to strong oscillations, whereas in Figure 7.4(b) we plot the solution with the full upwind scheme. Figure 7.3(b) plots the contour lines of the solution from the uniform generated mesh with 196008 elements. We see a significant transport from the square f = 5 in the direction of the convection vector b. Since u0 = 0 the contour lines are continuous on the boundary, which can be seen on (0, 0) to (0.25, 0). On (0, 0) to (0, −0.25) and (0.25, 0.25) to (0.25, 0) the resolution is not high enough. We remark that the exterior problem has only a diffusion term, which can be seen on the circular contour lines. 8. Conclusions. In this work a diffusion convection reaction problem was approximated by the finite volume element method, whereas the Laplace problem was simultaneously solved by the boundary element method in the corresponding exterior domain. Both are connected by appropriate transmission conditions on the coupling boundary. Our approach is very attractive in fluid dynamics with a dominated convection term in an interior domain, where one can use an upwind scheme, and for a diffusion process in a possibly unbounded exterior domain. We proved uniqueness and existence for both, our model problem and a corresponding discrete solution gained by a FVEM-BEM coupling, also with an upwind scheme for the convection part. The convergence and an a priori result of the proposed coupling were confirmed theoretical and in two numerical examples also with a special focus on convection dominated Coupling FVEM and BEM 21 problems. A forthcoming work will be to develop and prove an a posteriori error es- timator which is appropriate for adaptive mesh refinement. Also nonlinear diffusions coefficients are not considered here since they need more investigations. Acknowledgements. I gratefully acknowledge the support of Prof. Dr. Stefan A. Funken (University of Ulm, Germany) in implementation aspects. I also appreciate the helpful comments and suggestions of Prof. Dr. Dirk Praetorius (Vienna University of Technology, Austria) to prove Corollary 5.4. This work was partially supported by a postgraduate scholarship issued by the federal state Baden-Württemberg, Germany, and by DOE award DE-SC0001658.

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