Thesis Defense: Hybridizable Discontinuous Galerkin Methods in Finite Element Exterior Calculus

Abstract: We discuss hybridizable discontinuous Galerkin (HDG) methods for the Hodge-Dirac and Hodge-Laplace problems in finite element exterior calculus (FEEC). While most analysis in FEEC has been derived from Arnold, Falk, and Winther's commuting projections, we develop a novel approach drawing on Castillo, Cockburn, Perugia, and Schötzau's analysis of LDG methods. We show that HDG methods for both problems satisfy existence/uniqueness for both the local and global solvers, and for the Hodge-Laplace problem, we show that an alternative hybridization yields an SPD condensed system for the global variables. We analyze both methods by deriving equivalent un-hybridized formulations and obtain error results by applying the framework of Castillo et al., with much of the Hodge-Laplace analysis done by invoking the Hodge-Dirac results. For the Hodge-Dirac problem, optimal convergence is shown for all solution variables. For the Hodge-Laplace problem, optimal convergence for the solution variable u is shown, but only sub-optimal convergence in general for the auxiliary variables. However, optimal convergence is shown in all three variables for the n=2, k=1 case, representing an improvement on known results for the vector Laplacian. Numerical experiments support all results, and examples are shown for exact solutions with maximal or minimal regularity.

Faculty Advisor: Ari Stern