A Unified Space-Time Variational Framework for Wave Equations on Differential Forms

Abstract

We revisit a standard space--time variational formulation of the scalar wave equation in the setting of differential forms. For this, we use as spatial operator the nonnegative, self-adjoint (relative) \emph{Hodge--Laplacian} on a bounded Lipschitz domain. Its spectral decomposition reduces the wave problem in any form degree to the same family of scalar ordinary differential equations that underlies existing space-time analysis. We illustrate how this can be used to transfer results from the scalar wave equation with Dirichlet boundary conditions to other wave-type problems. In particular, we connect the framework to perfectly electrically conducting Maxwell problems.