A Unified Discontinuous Petrov--Galerkin Method and Its Analysis for Friedrichs' Systems

Abstract

We propose a unified discontinuous Petrov--Galerkin (DPG) framework with optimal test functions for Friedrichs-like systems, which embrace a large class of elliptic, parabolic, and hyperbolic partial differential equations (PDEs). The well-posedness, i.e., existence, uniqueness, and stability, of the DPG solution is established on a single abstract DPG formulation, and two abstract DPG methods corresponding to two different, but equivalent, norms are devised. We then apply the single DPG framework to several linear(ized) PDEs including, but not limited to, scalar transport, Laplace, diffusion, convection-diffusion, convection-diffusion-reaction, linear(ized) continuum mechanics (e.g., linear(ized) elasticity, a version of linearized Navier--Stokes equations, etc.), time-domain acoustics, and a version of the Maxwell's equations. The results show that we not only recover several existing DPG methods, but also discover new DPG methods for both PDEs currently considered in the DPG community and new ones. As a direct consequence of the single abstract DPG framework, all of the resulting DPG methods are shown to be trivially well-posed. We show that the inf-sup constant of the abstract DPG equation is independent of the mesh and is the same order as that of the PDE counterpart.