Unstructured Geometric Multigrid in Two and Three Dimensions on Complex and Graded Meshes

TL;DR

This paper presents a simplified mesh coarsening algorithm for unstructured geometric multigrid methods applicable to complex and graded meshes, improving efficiency in finite element problems with irregular domains.

Contribution

A new, simplified topologically-motivated mesh coarsening algorithm tailored for unstructured multigrid hierarchies on complex meshes, with implementation insights and computational validation.

Findings

Effective coarsening on complex, graded meshes

Improved multigrid performance on adaptive finite element problems

Compatibility with non-quasi-uniform mesh quality criteria

Abstract

The use of multigrid and related preconditioners with the finite element method is often limited by the difficulty of applying the algorithm effectively to a problem, especially when the domain has a complex shape or adaptive refinement. We introduce a simplification of a general topologically-motivated mesh coarsening algorithm for use in creating hierarchies of meshes for geometric unstructured multigrid methods. The connections between the guarantees of this technique and the quality criteria necessary for multigrid methods for non-quasi-uniform problems are noted. The implementation details, in particular those related to coarsening, remeshing, and interpolation, are discussed. Computational tests on pathological test cases from adaptive finite element methods show the performance of the technique.