A posteriori error estimation and adaptivity for degenerate parabolic problems
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- by R. H. Nochetto, A. Schmidt and C. Verdi PDF
- Math. Comp. 69 (2000), 1-24 Request permission
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.References
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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
- MR Author ID: 131850
- Email: rhn@math.umd.edu
- A. Schmidt
- Affiliation: Institut für Angewandte Mathematik, Universität Freiburg, 79106 Freiburg, Germany
- Email: alfred@mathematik.uni-freiburg.de
- C. Verdi
- Affiliation: Dipartimento di Matematica, Università di Milano, 20133 Milano, Italy
- Email: verdi@paola.mat.unimi.it
- 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.
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp. 69 (2000), 1-24
- MSC (1991): Primary 65N15, 65N30, 65N50, 80A22, 35K65, 35R35
- DOI: https://doi.org/10.1090/S0025-5718-99-01097-2
- MathSciNet review: 1648399