The locally adapted patch finite element method for interface problems on triangular meshes

TL;DR

This paper introduces a locally adapted finite element method for interface problems on triangular meshes, achieving optimal convergence by modifying macro elements to accurately represent interfaces within the mesh.

Contribution

The paper proposes a novel locally adapted parametric finite element method that handles interfaces independently of the mesh, ensuring optimal convergence for elliptic interface problems.

Findings

Achieves optimal convergence rates for elliptic interface problems.

Macro elements can be cut by interfaces and still maintain accuracy.

Method is well-suited for distributed computing architectures.

Abstract

We present a locally adapted parametric finite element method for interface problems. For this adapted finite element method we show optimal convergence for elliptic interface problems with a discontinuous diffusion parameter. The method is based on the adaption of macro elements where a local basis represents the interface. The macro elements are independent of the interface and can be cut by the interface. A macro element which is a triangle in the triangulation is divided into four subtriangles. On these subtriangles, the basis functions of the macro element are interpreted as linear functions. The position of the vertices of these subtriangles is determined by the location of the interface in the case a macro element is cut by the interface. Quadrature is performed on the subtriangles via transformations to a reference element. Due to the locality of the method, its use is well suited on distributed architectures.