  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 {\em infinite dimensional} mesh-dependent variational
formulation, in which besides the principal field also its trace on the mesh
skeleton is 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 states then 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.
