Numerical Convergence of Discrete Exterior Calculus on Arbitrary Surface Meshes

TL;DR

This paper demonstrates through numerical experiments that discrete exterior calculus (DEC) converges on arbitrary surface meshes, removing the need for Delaunay triangulations and simplifying mesh generation for PDE solutions.

Contribution

The study provides evidence that DEC converges on non-Delaunay meshes, challenging the assumption that special triangulations are necessary for accurate results.

Findings

DEC converges on various meshes including non-Delaunay.

Errors decrease as mesh refinement increases.

Signed diagonal Hodge star operator works effectively.

Abstract

Discrete exterior calculus (DEC) is a structure-preserving numerical framework for partial differential equations solution, particularly suitable for simplicial meshes. A longstanding and widespread assumption has been that DEC requires special (Delaunay) triangulations, which complicated the mesh generation process especially on curved surfaces. This paper presents numerical evidences demonstrating that this restriction is unnecessary. Convergence experiments are carried out for various physical problems using both Delaunay and non-Delaunay triangulations. Signed diagonal definition for the key DEC operator (Hodge star) is adopted. The errors converge as expected for all considered meshes and experiments. This relieves the DEC paradigm from unnecessary triangulation limitation.