Recovery-Based Error Estimators for Diffusion Problems: Explicit Formulas

TL;DR

This paper develops explicit recovery-based a posteriori error estimators for finite element methods solving diffusion problems with complex coefficients, improving robustness and computational efficiency.

Contribution

It introduces fully explicit error estimators for various finite element methods, enhancing robustness for interface problems with piecewise constant coefficients.

Findings

Estimators are robust with respect to jump sizes in coefficients.

Numerical experiments confirm theoretical robustness.

Explicit formulas improve computational efficiency.

Abstract

We introduced and analyzed robust recovery-based a posteriori error estimators for various lower order finite element approximations to interface problems in [9, 10], where the recoveries of the flux and/or gradient are implicit (i.e., requiring solutions of global problems with mass matrices). In this paper, we develop fully explicit recovery-based error estimators for lower order conforming, mixed, and non- conforming finite element approximations to diffusion problems with full coefficient tensor. When the diffusion coefficient is piecewise constant scalar and its distribution is local quasi-monotone, it is shown theoretically that the estimators developed in this paper are robust with respect to the size of jumps. Numerical experiments are also performed to support the theoretical results.