Stabilized finite element methods with shock capturing for advection–diffusion problems

Abstract Stabilized FEM of streamline-diffusion type for advection–diffusion problems may exhibit local oscillations in crosswind direction(s). As a remedy, a shock-capturing variant of such stabilized schemes is considered as an additional consistent (but nonlinear) stabilization. We prove existence of discrete solutions. Then we present some a priori and a posteriori estimates. Finally we address the efficient solution of the arising nonlinear discrete problems.

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