On the stability of DPG formulations of transport equations

In this paper we formulate and analyze a Discontinuous Petrov- Galerkin formulation of linear transport equations with variable convection fields. We show that a corresponding infinite dimensional mesh-dependent variational formulation, in which besides the principal field its trace on the mesh skeleton is also an unknown, is uniformly stable with respect to the mesh, where the test space is a certain product space over the underlying domain partition. Our main result then states the following. For piecewise polynomial trial spaces of degree m, we show under mild assumptions on the convection field that piecewise polynomial test spaces of degree m+1 over a refinement of the primal partition with uniformly bounded refinement depth give rise to uniformly (with respect to the mesh size) stable Petrov-Galerkin discretizations. The partitions are required to be shape regular but need not be quasi-uniform. An important startup ingredient is that for a constant convection field one can identify the exact optimal test functions with respect to a suitably modified but uniformly equivalent broken test space norm as piecewise polynomials. These test functions are then varied towards simpler and stably computable near-optimal test functions for which the above result is derived via a perturbation analysis. We conclude indicating some consequences of the results that will be treated in forthcoming work.