A class of discontinuous Petrov–Galerkin methods. Part III: Adaptivity

We continue our theoretical and numerical study on the Discontinuous Petrov–Galerkin method with optimal test functions in context of 1D and 2D convection-dominated diffusion problems and hp-adaptivity. With a proper choice of the norm for the test space, we prove robustness (uniform stability with respect to the diffusion parameter) and mesh-independence of the energy norm of the FE error for the 1D problem. With hp-adaptivity and a proper scaling of the norms for the test functions, we establish new limits for solving convection-dominated diffusion problems numerically: ϵ = 10 − 11 for 1D and ϵ = 10 − 7 for 2D problems. The adaptive process is fully automatic and starts with a mesh consisting of few elements only.