Adaptive finite elements with anisotropic meshes

TL;DR

This paper demonstrates that anisotropic mesh adaptation in finite element methods significantly improves solution accuracy for complex problems, with manageable conditioning of the resulting linear systems.

Contribution

It provides a comparative numerical study showing the advantages of anisotropic meshes over isotropic and quasi-uniform meshes in finite element analysis.

Findings

Anisotropic meshes yield higher accuracy in complex problems.

Conditioning of linear systems with adaptive anisotropic meshes is better than expected.

Numerical experiments confirm the effectiveness of anisotropic mesh adaptation.

Abstract

The paper presents a numerical study for the finite element method with anisotropic meshes. We compare the accuracy of the numerical solutions on quasi-uniform, isotropic, and anisotropic meshes for a test problem which combines several difficulties of a corner singularity, a peak, a boundary layer, and a wavefront. Numerical experiment clearly shows the advantage of anisotropic mesh adaptation. The conditioning of the resulting linear equation system is addressed as well. In particular, it is shown that the conditioning with adaptive anisotropic meshes is not as bad as generally assumed.