Breaking spaces and forms for the DPG method and applications including Maxwell equations

TL;DR

This paper introduces a novel approach to DPG methods using broken spaces derived from unbroken Sobolev spaces, enabling simplified stability and error analysis for Maxwell equations and other PDEs.

Contribution

It characterizes interface spaces connecting broken and unbroken spaces, simplifying stability analysis and providing comprehensive error estimates for Maxwell equations within the DPG framework.

Findings

Complete error analysis for Maxwell equations with perfect electric boundary conditions.

Simplified stability proofs for various Maxwell problem formulations.

Reliability and efficiency estimates for error indicators.

Abstract

Discontinuous Petrov Galerkin (DPG) methods are made easily implementable using `broken' test spaces, i.e., spaces of functions with no continuity constraints across mesh element interfaces. Broken spaces derivable from a standard exact sequence of first order (unbroken) Sobolev spaces are of particular interest. A characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided. Stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces. This technique is used to provide a complete error analysis of DPG methods for Maxwell equations with perfect electric boundary conditions. The technique also permits considerable simplifications of previous analyses of DPG methods for other equations. Reliability and efficiency estimates for an error indicator also follow. Finally, the equivalence of stability for various formulations of the same Maxwell problem is proved, including the strong form, the ultraweak form, and a spectrum of forms in between.