Hierarchical a posteriori error estimation of Bank-Weiser type in the FEniCS Project

TL;DR

This paper introduces a new implementation of hierarchical Bank-Weiser type a posteriori error estimators within the FEniCS Project, demonstrating their effectiveness in adaptive mesh refinement and mixed elasticity problems.

Contribution

It presents a novel, practical implementation of Bank-Weiser estimators in a high-level finite element framework with automatic code generation.

Findings

Effective in goal-oriented adaptive mesh refinement

Improves mixed approximation accuracy in elasticity problems

Comparable or superior to other estimators in tests

Abstract

In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp.283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element mesh. Despite the promise of Bank-Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, they have seen little use in practical computational problems. The focus of this contribution is to describe a novel implementation of hierarchical estimators of the Bank-Weiser type in a modern high-level finite element software with automatic code generation capabilities. We show how to use the estimator to drive (goal-oriented) adaptive mesh refinement and to mixed approximations of the nearly-incompressible elasticity problems. We provide comparisons with various other used estimators. An open-source implementation based on the FEniCS Project finite element software is provided as supplementary material.