A UNIFIED ERROR ANALYSIS OF HYBRIDIZABLE DISCONTINUOUS GALERKIN METHODS FOR THE STATIC MAXWELL EQUATIONS

摘要

We propose a framework that allows us to analyze different variants of hybridizable discontinuous Galerkin (HDG) methods for the static Maxwell equations using one simple analysis. It reduces all the work to the construction of projections that best fit the structures of the approximation spaces. As applications, we analyze four variants of HDG methods (denoted by B, H, B+, H+), where two of them are known (variants H, B+) and the other two are new (variants H+, B). Under certain regularity assumptions, we show that all four variants are optimally convergent and that variants B+ and H+ achieve superconvergence without postprocessing. For the two known variants, we prove their optimal convergence under weaker requirements of the meshes and the stabilization functions thanks to the new analysis techniques being introduced. For solution with low-regularity, we give an analysis to these methods and investigate the effect of different stabilization functions on the convergence. At the end, we provide numerical experiments to support the analysis.