Convergence of discrete exterior calculus approximations for Poisson problems

Abstract

Discrete exterior calculus (DEC) is a framework for constructing discrete versions of exterior differential calculus objects, widely used in computer graphics, computational topology, and discretizations of the Hodge-Laplace operator. However, a rigorous convergence analysis of DEC has always been lacking. 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. Numerical experiments show that the convergence is in fact of second order when the solution is sufficiently regular.