Robust DPG Method for Convection-Dominated Diffusion Problems

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We propose and analyze a discontinuous Petrov--Galerkin (DPG) method for convection-dominated diffusion problems that provides robust $L^2$ error estimates for the field variables which are quasi-optimal in the energy norm. A key feature of the method is to construct test functions defined by a variational formulation with bilinear form (test norm) specifically designed for the goal of robustness. A main theoretical ingredient is a stability analysis of the adjoint problem. Numerical experiments underline our theoretical results and, in particular, confirm robustness of the DPG method for well-chosen test norms.