Finite Element Methods for Interface Problems: Robust and Local Optimal A Priori Error Estimates

TL;DR

This paper develops robust and locally optimal a priori error estimates for various finite element methods applied to elliptic interface problems, without assumptions on the diffusion coefficient distribution.

Contribution

It provides the first robust, optimal a priori error estimates for Crouzeix-Raviart, Raviart-Thomas, and discontinuous Galerkin methods on interface problems.

Findings

Error estimates are robust with respect to the diffusion coefficient.

Estimates are optimal concerning local regularity of the solution.

No assumptions on the diffusion coefficient distribution are needed.

Abstract

For elliptic interface problems in two- and three-dimensions, this paper establishes a priori error estimates for Crouzeix-Raviart nonconforming, Raviart-Thomas mixed, and discontinuous Galerkin finite element approximations. These estimates are robust with respect to the diffusion coefficient and optimal with respect to local regularity of the solution. Moreover, we obtain these estimates with no assumption on the distribution of the diffusion coefficient.