On the non-symmetric coupling method for parabolic-elliptic interface problems
This work provides rigorous numerical analysis for a coupling method applied to a class of interface problems, addressing gaps in theory for non-smooth interfaces.
The authors prove well-posedness and quasi-optimality for the non-symmetric coupling method applied to parabolic-elliptic interface problems, deriving optimal-order error estimates for both semi-discrete and fully discrete schemes under minimal regularity assumptions, without relying on duality arguments.
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.