Local Coarsening Algorithms on Adaptively Refined Meshes in 2D and Their Efficient Implementation in MATLAB

TL;DR

This paper introduces efficient local coarsening algorithms for adaptively refined triangular and quadrilateral meshes in 2D, implemented in MATLAB, extending the ameshref toolbox with implicit refinement history handling.

Contribution

It presents novel coarsening algorithms that operate without explicit refinement history, leveraging data structures for efficiency and flexibility in MATLAB implementations.

Findings

Algorithms guarantee mesh quality properties.

Implementation in MATLAB is efficient and practical.

Numerical examples demonstrate effectiveness.

Abstract

Adaptive meshing includes local refinement as well as coarsening of meshes. Typically, coarsening algorithms are based on an explicit refinement history. In this work, we deal with local coarsening algorithms that build on the refinement strategies for triangular and quadrilateral meshes implemented in the ameshref package (Funken and Schmidt 2018, 2019). The ameshref package is a MATLAB-toolbox for research and teaching purposes which offers the user a certain flexibility in the REFINE step of an adaptive finite element method but can also be used in other contexts like computer graphics. This toolbox is now be extended by the coarsening option. In ameshref, no explicit information about the refinement process is stored, but is instead implicit in the data structure. In this work, we present coarsening algorithms that use easy-to-verify criteria to coarsen adaptively generated meshes by exploiting the data structure. Thereby, the desired properties are guaranteed and computational efficiency is maintained. A MATLAB implementation and some numerical examples are discussed in this work and are included in full in the toolbox ameshcoars (Funken and Schmidt 2020).