A framework for robust a posteriori error control in unsteady nonlinear advection-diffusion problems
Résumé
We derive a framework for a posteriori error estimates in unsteady, nonlinear, possibly degenerate, advection-diffusion problems. Our estimators are based on space-time equilibrated flux reconstruction and are locally computable. They are derived for the error measured in a space-time mesh-dependent dual norm stemming from the problem and meshes at hand augmented by a jump-based contribution measuring possible nonconformities in space. Owing to this choice, a guaranteed upper bound, as well as global efficiency and robustness with respect to all model (e.g., nonlinearities, advection dominance, final time) and discretization parameters are achieved. Local-in-time and in-space efficiency is also shown for a localized upper bound of the error measure. In order to apply the framework to a given numerical method, two simple conditions, local space-time mass conservation and an approximation property of the reconstructed fluxes, need to be verified. We show how to do this for the discontinuous Galerkin method in space and the Crank-Nicolson scheme in time. Numerical experiments illustrate the theory.
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