Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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
Full-text PDF Free Access

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 $C^{0}$ 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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65N15, 65N30, 65N50, 80A22, 35K65, 35R35

Retrieve articles in all journals with MSC (1991): 65N15, 65N30, 65N50, 80A22, 35K65, 35R35


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: http://dx.doi.org/10.1090/S0025-5718-99-01097-2
PII: S 0025-5718(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