Two explicit error representation formulas are derived for degenerate parabolic PDE, which are based on evaluating a parabolic residual in negative norms. The resulting upper bounds are valid for any numerical method, and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient. They are applied to a practical space-time discretization consisting of C0 piecewise linear finite elements over highly graded unstructured meshes, and backward finite differences with varying time-steps. Two rigorous a posteriori error estimates are derived for this scheme, and used in designing an efficient adaptive algorithm, which equidistributes space and time discretization errors via refinement/coarsening. Several simulations compare both estimators, illustrate the overall scheme efficiency (including various solvers, hierarchical data structures, error estimation, and mesh modification), and reveal reliability and potentials of the adaptive method. Finally convergence is shown under suitable coarsening restrictions.
This is joint work with A. Schmidt and C. Verdi.
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