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.

**[1]**I. Babu\v{s}ka and W.C. Rheinboldt,*Error estimates for adaptive finite element computations*, SIAM J. Numer. Anal.**15**(1978), 736-754. MR**58:3400****[2]**E. Bänsch,*Local mesh refinement in 2 and 3 dimensions*, IMPACT Comput. Sci. Engrg.**3**(1991), 181-191. MR**92h:65150****[3]**P.G. Ciarlet,*The Finite Element Method for Elliptic Problems*, North-Holland, Amsterdam, 1978. MR**58:25001****[4]**Ph. Clément,*Approximation by finite element functions using local regularization*, RAIRO Anal. Numér.**9**(1975), 77-84. MR**53:4569****[5]**K. Eriksson and C. Johnson,*Adaptive finite element methods for parabolic problems I: a linear model problem*, SIAM J. Numer. Anal.**28**(1991), 43-77. MR**91m:65274****[6]**-,*Adaptive finite element methods for parabolic problems II: optimal error estimates in and*, SIAM J. Numer. Anal.**32**(1995), 706-740. MR**96c:65162****[7]**-,*Adaptive finite element methods for parabolic problems IV: nonlinear problems*, SIAM J. Numer. Anal.**32**(1995), 1750-1763. MR**96i:65082****[8]**A. Friedman,*Variational Principles and Free Boundary Problems*, Wiley, New York, 1982. MR**84e:35153****[9]**O.A. Ladyzenskaja, V. Solonnikov, and N. Ural'ceva,*Linear and Quasilinear Equations of Parabolic Type*, AMS, Providence, 1968. MR**39:3159****[10]**R.H. Nochetto,*Error estimates for multidimensional singular parabolic problems*, Japan J. Appl. Math.**4**(1987), 111-138. MR**89c:65107****[11]**R.H. Nochetto, M. Paolini, and C. Verdi,*An adaptive finite elements method for two-phase Stefan problems in two space dimensions. Part I: stability and error estimates. Supplement*, Math. Comp.**57**(1991), 73-108, S1-S11. MR**92a:65322****[12]**-,*An adaptive finite elements method for two-phase Stefan problems in two space dimensions. Part II: implementation and numerical experiments*, SIAM J. Sci. Statist. Comput.**12**(1991), 1207-1244. MR**92f:65138****[13]**R.H. Nochetto, A. Schmidt, and C. Verdi,*Adapting meshes and time-steps for phase change problems*, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. (9) Mat. Appl.**8**(1997), 273-292. CMP**98:15****[14]**-,*Adaptive solution of phase change problems over unstructured tetrahedral meshes*, Grid Generation and Adaptive Algorithms (M. Luskin*et al.*eds.), Springer Verlag, New York (to appear).**[15]**P.A. Raviart,*The use of numerical integration in finite element methods for solving parabolic equations*, Topics in Numerical Analysis (J.J.H. Miller ed.), Academic Press, London, 1973, pp. 233-264. MR**49:10164**

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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