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

摘要

Discontinuous Petrov Galerkin (DPG) methods are made easily implementableusing `broken' test spaces, i.e., spaces of functions with no continuityconstraints across mesh element interfaces. Broken spaces derivable from astandard exact sequence of first order (unbroken) Sobolev spaces are ofparticular interest. A characterization of interface spaces that connect thebroken spaces to their unbroken counterparts is provided. Stability of certainformulations using the broken spaces can be derived from the stability ofanalogues that use unbroken spaces. This technique is used to provide acomplete error analysis of DPG methods for Maxwell equations with perfectelectric boundary conditions. The technique also permits considerablesimplifications 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 sameMaxwell problem is proved, including the strong form, the ultraweak form, and aspectrum of forms in between.