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A posteriori error estimation and adaptivity
for degenerate parabolic problems

Authors: R. H. Nochetto, A. Schmidt and C. Verdi
Journal: Math. Comp. 69 (2000), 1-24
MSC (1991): Primary 65N15, 65N30, 65N50, 80A22, 35K65, 35R35
Published electronically: August 24, 1999
MathSciNet review: 1648399
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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 $C^{0}$ 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.

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Additional Information

R. H. Nochetto
Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA

A. Schmidt
Affiliation: Institut für Angewandte Mathematik, Universität Freiburg, 79106 Freiburg, Germany

C. Verdi
Affiliation: Dipartimento di Matematica, Università di Milano, 20133 Milano, Italy

Keywords: Degenerate parabolic equations, Stefan problem, finite elements, parabolic duality, a posteriori estimates, adaptivity
Received by editor(s): June 9, 1997
Published electronically: August 24, 1999
Additional Notes: This work was partially supported by NSF Grants DMS-9305935 and DMS-9623394, EU Grant HCM “Phase Transitions and Surface Tension”, MURST, and CNR Contract 95.00735.01.
Article copyright: © Copyright 1999 American Mathematical Society