Coupling of the Finite Volume Method and the Boundary Element Method - Theory, Analysis, and Numerics

Abstract

We develop a discretization scheme for the coupling of the finite volume method and the boundary element method in two dimensions, which describes, for example, the transport of a concentration in a fluid. The discrete system maintains naturally local conservation. In a bounded interior domain we approximate a diffusion convection reaction problem either by the finite volume element method or by the cell-centered finite volume method, whereas in the corresponding exterior domain the Laplace problem is solved by the boundary element method. On the coupling boundary we have appropriate transmission conditions. A weighted upwind scheme guarantees the stability of the method also for convection dominated problems. We show existence and uniqueness of the continuous system and provide an a~priori analysis for the coupling with the finite volume element method. For both coupling systems we derive residual-based a~posteriori estimates, which give upper and lower bounds for the error between the exact solution and the approximate solution. These bounds measure the error in an energy (semi-) norm and are robust in the sense that they do not depend on the variation of the model data. The local contributions of the a~posteriori estimates are used to steer an adaptive mesh-refining algorithm. Numerical experiments show that our adaptive coupling is an efficient method for the numerical treatment of transmission problems, which exhibit local behavior.