A posteriori error estimation and adaptivity for degenerate parabolic problems

Abstract

Two explicit error representation formulas are derived for degenerateparabolic PDEs, which are based on evaluating a parabolic residualin negative norms. The resulting upper bounds are valid for any numericalmethod, and rely on regularity properties of solutions of a dualparabolic problem in nondivergence form with vanishing diffusioncoefficient. They are applied to a practical space-time discretizationconsisting of C0 piecewise linear finite elements over highly gradedunstructured meshes, and backward finite differences with varyingtime-steps. Two rigorous a posteriori error estimates are derivedfor this scheme, and used in designing an efficient adaptive algorithm,which equidistributes space and time discretization errors via refinement/coarsening.A simulation finally compares the behavior of the rigorous a posteriorierror estimators with a heuristic approach, and hints at the potentialsand reliability of the proposed method.