An upwind-mixed method for advection-diffusion problems with static condensation

In this work we consider the dual-primal Discontinuous Petrov-Galerkin (DPG) method for the advection-diffusion model problem. Since in the DPG method both mixed internal variables are discontinuous, a static condensation procedure can be carried out, leading to a single-field nonconforming discretization scheme. For this latter formulation, we propose a flux-upwind stabilization technique to deal with the advection-dominated case. The resulting scheme is conservative and satisfies a discrete maximum principle under standard geometrical assumptions on the computational grid. Convergence of the statically condensed formulation is established in a discrete H-1-norm, and the numerical performance of the scheme is validated on benchmark problems with sharp internal and boundary layers.