Stable non-symmetric coupling of the finite volume and the boundary element method for convection-dominated parabolic-elliptic interface problems
This work offers a numerically stable and conservative method for solving convection-dominated parabolic-elliptic interface problems, which is relevant for applications in electrical engineering and fluid mechanics.
The authors propose a non-symmetric coupling of the finite volume method (FVM) and the boundary element method (BEM) for convection-dominated parabolic-elliptic interface problems, addressing conservation and stability issues of FEM. They provide convergence and error estimates for semi-discrete and fully discrete schemes, with numerical examples confirming the theory.
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.