Higher-order finite element methods for elliptic problems with interfaces

TL;DR

This paper introduces higher-order finite element methods with correction terms for solving elliptic interface problems, achieving optimal error estimates and extending to Stokes interface problems with proven convergence.

Contribution

It develops a novel correction-based finite element approach for interface problems, providing optimal error estimates and extending to Stokes problems.

Findings

Optimal error estimates in maximum norms for the proposed methods

Extension of the method to Stokes interface problems with optimal convergence

Effective correction terms improve solution accuracy near interfaces

Abstract

We present higher-order piecewise continuous finite element methods for solving a class of interface problems in two dimensions. The method is based on correction terms added to the right-hand side in the standard variational formulation of the problem. We prove optimal error estimates of the methods on general quasi-uniform and shape regular meshes in maximum norms. In addition, we apply the method to a Stokes interface problem, adding correction terms for the velocity and the pressure, obtaining optimal convergence results.