Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions

TL;DR

This paper establishes the convergence and stability of discrete exterior calculus solutions for the Hodge-Laplace problem in two dimensions, relating DEC to finite element exterior calculus and relaxing mesh conditions.

Contribution

It proves convergence and stability of DEC for 2D Hodge-Laplace problems by linking DEC to FEEC and relaxing mesh regularity conditions.

Findings

DEC solutions are convergent and stable on Delaunay and shape regular meshes.

DEC and FEEC norms are equivalent under certain geometric conditions.

Stability and convergence are achieved with relaxed mesh conditions.

Abstract

We prove convergence and stability of the discrete exterior calculus (DEC) solutions for the Hodge-Laplace problems in two dimensions for families of meshes that are non-degenerate Delaunay and shape regular. We do this by relating the DEC solutions to the lowest order finite element exterior calculus (FEEC) solutions. A Poincar\'e inequality and a discrete inf-sup condition for DEC are part of this proof. We also prove that under appropriate geometric conditions on the mesh the DEC and FEEC norms are equivalent. Only one side of the norm equivalence is needed for proving stability and convergence and this allows us to relax the conditions on the meshes.