A pedagogical review on a posteriori error estimation in Finite Element computations

TL;DR

This review comprehensively discusses a posteriori error estimation techniques in Finite Element Methods, focusing on equilibrium-based methods for linear elliptic problems, and covers various approaches including recovery, residual, and duality methods.

Contribution

It unifies and explains a wide range of a posteriori error estimation methods centered on the equilibrium concept for linear elliptic problems.

Findings

Error bounds are fully computable and mathematically certified.

Recovery, residual, and duality-based methods effectively estimate solution errors.

Numerical examples illustrate the application to 3D linear elasticity problems.

Abstract

This article is a review on basic concepts and tools devoted to a posteriori error estimation for problems solved with the Finite Element Method. For the sake of simplicity and clarity, we mostly focus on linear elliptic diffusion problems, approximated by a conforming numerical discretization. The review mainly aims at presenting in a unified manner a large set of powerful verification methods, around the concept of equilibrium. Methods based on that concept provide error bounds that are fully computable and mathematically certified. We discuss recovery methods, residual methods, and duality-based methods for the estimation of the whole solution error (i.e. the error in energy norm), as well as goal-oriented error estimation (to assess the error on specific quantities of interest). We briefly survey the possible extensions to non-conforming numerical methods, as well as more complex (e.g. nonlinear or time-dependent) problems. We also provide some illustrating numerical examples on a linear elasticity problem in 3D.