On the non-symmetric coupling method for parabolic-elliptic interface problems

TL;DR

This paper analyzes the non-symmetric coupling method for parabolic-elliptic interface problems, establishing well-posedness, quasi-optimality, and optimal error estimates for semi- and fully discrete schemes, supported by numerical examples.

Contribution

It provides the first comprehensive analysis of the non-symmetric coupling method for these problems, including error estimates under minimal regularity assumptions.

Findings

Proved well-posedness of the coupling formulation.

Established quasi-optimality and optimal error estimates for semi- and fully discrete schemes.

Numerical examples confirm theoretical results.

Abstract

We consider the numerical approximation of parabolic-elliptic interface problems by the non-symmetric coupling method of MacCamy and Suri [Quart. Appl. Math., 44 (1987), pp. 675--690]. We establish well-posedness of this formulation for problems with non-smooth interfaces and prove quasi-optimality for a class of conforming Galerkin approximations in space. Therefore, error estimates with optimal order can be deduced for the semi-discretization in space by appropriate finite and boundary elements. Moreover, we investigate the subsequent discretization in time by a variant of the implicit Euler method. As for the semi-discretization, we establish well-posedness and quasi-optimality for the fully discrete scheme under minimal regularity assumptions on the solution. Error estimates with optimal order follow again directly. Our analysis is based on estimates in appropriate energy norms. Thus, we do not use duality arguments and corresponding estimates for an elliptic projection which are not available for the non-symmetric coupling method. Additionally, we provide again error estimates under minimal regularity assumptions. Some numerical examples illustrate our theoretical results.