On the implementation of a locally modified finite element method for interface problems in deal.II

TL;DR

This paper presents a simple, locally modified finite element method for interface problems that accurately resolves weak discontinuities without changing the global mesh, implemented in deal.II and validated through numerical examples.

Contribution

It introduces a fitted finite element approach with local degrees of freedom incorporated parametrically, avoiding mesh construction while resolving interfaces accurately.

Findings

Effective resolution of weak discontinuities demonstrated

Implementation details provided for deal.II library

Numerical examples validate the approach's performance

Abstract

In this work, we describe a simple finite element approach that is able to resolve weak discontinuities in interface problems accurately. The approach is based on a fixed patch mesh consisting of quadrilaterals, that will stay unchanged independent of the position of the interface. Inside the patches we refine once more, either in eight triangles or in four quadrilaterals, in such a way that the interface is locally resolved. The resulting finite element approach can be considered a fitted finite element approach. In our practical implementation, we do not construct this fitted mesh, however. Instead, the local degrees of freedom are included in a parametric way in the finite element space, or to be more precise in the local mappings between a reference patch and the physical patches. We describe the implementation in the open source C++ finite element library deal.II in detail and present two numerical examples to illustrate the performance of the approach. Finally, detailed studies of the behavior of iterative linear solvers complement this work.