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

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 Poincaré 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.

Section 2.2: Hodge-Laplace problem is now given in weak form. Remark 4.10: points out connection to Holst and Stern (2012). Remark 5.5: Points out how to obtain a slightly stronger version of Theorem 5.4. Proof of Theorem 6.1: the condition "for sufficiently small h" is removed. Equation numbering is now section specific