A Framework for Analysis of DEC Approximations to Hodge-Laplacian Problems using Generalized Whitney Forms
For researchers in numerical analysis and computational geometry, it provides a rigorous theoretical foundation for DEC methods, enabling convergence guarantees and error analysis.
This paper establishes a framework linking Discrete Exterior Calculus (DEC) to Finite Element Exterior Calculus (FEEC), proving convergence with rates for the Hodge-Laplacian problem on well-centered meshes and explaining superconvergence phenomena.
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.