A unified error analysis of HDG methods for the static Maxwell equations

TL;DR

This paper introduces a unified framework for analyzing various HDG methods applied to static Maxwell equations, demonstrating optimal convergence and superconvergence properties through a simplified projection-based approach.

Contribution

It develops a new, unified analysis technique for HDG methods, including two novel variants, showing their optimal and superconvergence properties under broader conditions.

Findings

All four HDG variants are optimally convergent.

Variants B+ and H+ achieve superconvergence without post-processing.

The analysis extends to low-regularity solutions and different stabilization functions.

Abstract

We propose a framework that allows us to analyze different variants of 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 assumption, we show that all the four variants are optimally convergent and that variants B+ and H+ achieve superconvergence without post-processing. 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.