Finite Element Methods for Interface Problems: Robust Residual-Based A Posteriori Error Estimates

TL;DR

This paper develops robust residual-based a posteriori error estimators for various finite element methods applied to elliptic interface problems, ensuring reliability independent of coefficient jumps and distribution.

Contribution

It introduces new error bounds for multiple finite element methods that are independent of diffusion coefficient jumps and distribution, enhancing robustness.

Findings

Reliability bounds for conforming and Raviart-Thomas elements in 2D.

Reliability bounds for Crouzeix-Raviart and discontinuous Galerkin elements in 2D and 3D.

Constants in estimates are independent of diffusion coefficient jumps.

Abstract

For elliptic interface problems, this paper studies residual-based a posteriori error estimations for various finite element approximations. For the conforming and the Raviart-Thomas mixed elements in two-dimension and for the Crouzeix-Raviart nonconforming and the discontinuous Galerkin elements in both two- and three-dimensions, the global reliability bounds are established with constants independent of the jump of the diffusion coefficient. Moreover, we obtain these estimates with no assumption on the distribution of the diffusion coefficient.