A Study on Using Hierarchical Basis Error Estimates in Anisotropic Mesh Adaptation for the Finite Element Method

TL;DR

This paper evaluates a hierarchical basis error estimator for anisotropic mesh adaptation in finite element methods, demonstrating its effectiveness across various applications and its advantages in handling complex solution features.

Contribution

It introduces and assesses a hierarchical basis error estimator for anisotropic mesh adaptation, showing its competitive performance and benefits over traditional Hessian-based methods.

Findings

Performs well across different applications

Provides better adaptation for gradient jumps and boundary layers

Generates meshes comparable to theoretical optima in singular problems

Abstract

A common approach for generating an anisotropic mesh is the M-uniform mesh approach where an adaptive mesh is generated as a uniform one in the metric specified by a given tensor M. A key component is the determination of an appropriate metric which is often based on some type of Hessian recovery. Recently, the use of a global hierarchical basis error estimator was proposed for the development of an anisotropic metric tensor for the adaptive finite element solution. This study discusses the use of this method for a selection of different applications. Numerical results show that the method performs well and is comparable with existing metric tensors based on Hessian recovery. Also, it can provide even better adaptation to the solution if applied to problems with gradient jumps and steep boundary layers. For the Poisson problem in a domain with a corner singularity, the new method provides meshes that are fully comparable to the theoretically optimal meshes.