Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions
This work addresses theoretical guarantees for numerical methods in computational geometry and PDEs, but it is incremental as it builds on existing FEEC frameworks.
The paper tackled the problem of proving convergence and stability for discrete exterior calculus (DEC) solutions to Hodge-Laplace problems in two dimensions, achieving this by relating DEC to finite element exterior calculus (FEEC) and establishing norm equivalence under specific mesh conditions.
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.