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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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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


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
  • MR Author ID: 131850
  • Email:
  • A. Schmidt
  • Affiliation: Institut für Angewandte Mathematik,  Universität Freiburg,  79106 Freiburg, Germany
  • Email:
  • C. Verdi
  • Affiliation: Dipartimento di Matematica, Università di Milano, 20133 Milano, Italy
  • Email:
  • 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:
  • MathSciNet review: 1648399