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 prob- lems. 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 dif- ferential 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 el- ement method (FVEM). This method is a well-adapted method for the discretization of various partial differential equations in bounded domains and features local conser- vation 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 Calder´on 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 Poincar´e 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 func- tions. 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.