A locally modified second-order finite element method for interface problems and its implementation in 2 dimensions

TL;DR

This paper extends a locally modified finite element method to second order in 2D interface problems, providing theoretical error estimates and numerical validation for improved accuracy in resolving weak discontinuities.

Contribution

It introduces a second-order isoparametric extension of the method, ensuring curved edges do not cause degeneracy, with proven optimal error estimates.

Findings

Optimal a priori error estimates in $L^2$-norm and energy norm.

Numerical examples confirm theoretical error bounds.

Method effectively resolves weak discontinuities in interface problems.

Abstract

The locally modified finite element method, which is introduced in [Frei, Richter: SINUM 52(2014), p. 2315-2334], is a simple fitted finite element method that is able to resolve weak discontinuities in interface problems. The method is based on a fixed structured coarse mesh, which is then refined into sub-elements to resolve an interior interface. In this work, we extend the locally modified finite element method {in two space dimensions} to second order using an isoparametric approach in the interface elements. Thereby we need to take care that the resulting curved edges do not lead to degenerate sub-elements. We prove optimal a priori error estimates in the $L^2$-norm and in a discrete energy norm. Finally, we present numerical examples to substantiate the theoretical findings.