A Framework for Analysis of DEC Approximations to Hodge-Laplacian Problems using Generalized Whitney Forms

TL;DR

This paper introduces a framework connecting Discrete Exterior Calculus with Finite Element Exterior Calculus, enabling rigorous analysis and proof of convergence rates for Hodge-Laplacian problems using generalized Whitney forms.

Contribution

It establishes an equivalence between DEC and FEEC, allowing for convergence analysis and explanation of superconvergence phenomena in DEC schemes.

Findings

Proved convergence with explicit rates for Hodge-Laplacian in full $k$-form generality.

Numerical results confirm optimal convergence rates.

Framework explains superconvergence phenomena in DEC.

Abstract

We provide a framework for interpreting Discrete Exterior Calculus (DEC) numerical schemes in terms of Finite Element Exterior Calculus (FEEC). We demonstrate the equivalence of cochains on primal and dual meshes with Whitney and generalized Whitney forms which allows us to analyze DEC approximations using tools from FEEC. We demonstrate the applicability of our framework by rigorously proving convergence with rates for the Hodge-Laplacian problem in full $k$-form generality on well-centered meshes. We also provide numerical results illustrating optimality of our derived convergence rates. Moreover, we demonstrate how superconvergence phenomena can be explained in our framework with corresponding numerical results.