A posteriori error estimation and adaptivity for degenerate parabolic problems

Abstract. Two explicit error representation formulas are derived for degenerate parabolic PDEs, 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. A simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach, and hints at the potentials and reliability of the proposed method.

• Type: Journal
• Year: 2000
• Where: MOC
• Authors: Ricardo H. Nochetto, Alfred Schmidt, C. Verdi
• Keywords: Dual Parabolic Problem, Explicit Error Representation, MOC 2000, Posteriori Error