Convergence of Discrete Exterior Calculus Approximations for Poisson Problems

TL;DR

This paper provides a rigorous convergence analysis of discrete exterior calculus (DEC) for Poisson problems, establishing pointwise convergence in arbitrary dimensions and demonstrating second-order convergence in numerical experiments.

Contribution

It introduces a new framework for analyzing DEC convergence independently, extending results beyond two dimensions and the scalar Poisson problem.

Findings

DEC solutions converge pointwise to the exact solution at least linearly with mesh size

Numerical experiments show second-order convergence for sufficiently regular solutions

The framework helps explain convergence behavior and opens avenues for further research

Abstract

Discrete exterior calculus (DEC) is a framework for constructing discrete versions of exterior differential calculus objects, and is widely used in computer graphics, computational topology, and discretizations of the Hodge-Laplace operator and other related partial differential equations. However, a rigorous convergence analysis of DEC has always been lacking; as far as we are aware, the only convergence proof of DEC so far appeared is for the scalar Poisson problem in two dimensions, and it is based on reinterpreting the discretization as a finite element method. Moreover, even in two dimensions, there have been some puzzling numerical experiments reported in the literature, apparently suggesting that there is convergence without consistency. In this paper, we develop a general independent framework for analyzing issues such as convergence of DEC without relying on theories of other discretization methods, and demonstrate its usefulness by establishing convergence results for DEC beyond the Poisson problem in two dimensions. Namely, we prove that DEC solutions to the scalar Poisson problem in arbitrary dimensions converge pointwise to the exact solution at least linearly with respect to the mesh size. We illustrate the findings by various numerical experiments, which show that the convergence is in fact of second order when the solution is sufficiently regular. The problems of explaining the second order convergence, and of proving convergence for general p-forms remain open.