Stable non-symmetric coupling of the finite volume and the boundary element method for convection-dominated parabolic-elliptic interface problems

TL;DR

This paper develops a stable non-symmetric coupling of finite volume and boundary element methods for convection-dominated parabolic-elliptic interface problems, ensuring flux conservation and stability in numerical simulations.

Contribution

It introduces a novel FVM-BEM coupling with upwind stabilization and analyzes its convergence and error estimates under minimal regularity assumptions.

Findings

The proposed method is stable and convergent for convection-dominated problems.

Error estimates are established for semi-discrete and fully discrete schemes.

Numerical examples confirm theoretical results and demonstrate practical applicability.

Abstract

Many problems in electrical engineering or fluid mechanics can be modeled by parabolic-elliptic interface problems, where the domain for the exterior elliptic problem might be unbounded. A possibility to solve this class of problems numerically is the non-symmetric coupling of finite elements (FEM) and boundary elements (BEM) analyzed in [Egger, Erath, Schorr, arXiv:1711.08487, 2017]. If, for example, the interior problem represents a fluid, this method is not appropriate since FEM in general lacks conservation of numerical fluxes and in case of convection dominance also stability. A possible remedy to guarantee both is the use of the vertex-centered finite volume method (FVM) with an upwind stabilization option. Thus we propose a (non-symmetric) coupling of FVM and BEM for a semi-discretization of the underlying problem. For the subsequent time discretization we introduce two options: a variant of the backward Euler method which allows us to develop an analysis under minimal regularity assumptions and the classical backward Euler method. We analyze both, the semi-discrete and the fully discrete system, in terms of convergence and error estimates. Some numerical examples illustrate the theoretical findings and give some ideas for practical applications.