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

DOI:
https://doi.org/10.1090/S0025-5718-99-01097-2

Published electronically:
August 24, 1999

MathSciNet review:
1648399

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Abstract | References | Similar Articles | Additional Information

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 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

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

DOI:
https://doi.org/10.1090/S0025-5718-99-01097-2

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