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A posteriori error estimates for nonlinear problems.  -error estimates for finite element discretizations of parabolic equations
Author(s):
R.
Verfürth.
Journal:
Math. Comp.
67
(1998),
1335-1360.
MSC (1991):
Primary 65N30, 65N15, 65J15, 76D05
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Abstract:
Using the abstract framework of [9] we analyze a residual a posteriori error estimator for space-time finite element discretizations of quasilinear parabolic pdes. The estimator gives global upper and local lower bounds on the error of the numerical solution. The finite element discretizations in particular cover the so-called  -scheme, which includes the implicit and explicit Euler methods and the Crank-Nicholson scheme.
References:
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- R. A. Adams: Sobolev spaces. Academic Press, New York, 1975 MR 56:9247
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- H. Amann: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), Teubner, Stuttgart, 1993, pp. 9-126. MR 94m:35153
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- J. Baranger and H. El Amri: Estimateurs a posteriori d'erreur pour le calcul adaptif d'écoulements quasi-Newtoniens. RAIRO Modél Math. Anal. Numer. 25, 31 - 48 (1991) MR 91m:76070
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- K. Eriksson and C. Johnson: Adaptive finite element methods for parabolic problems IV. Nonlinear problems. SIAM J. Numer Anal. 32 (1995), 1729-1749. MR 96i:65081
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- -, Adaptive finite element methods for parabolic problems V. Long-time integration. SIAM J. Numer. Anal. 32 (1995), 1750-1763 MR 96i:65082
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- R. Verfürth: A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comput. 62 (206), 445 - 475 (1994) MR 94j:65136
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- -, A posteriori error estimates for nonlinear problems. Finite element discretizations of parabolic equations. Report 180, Ruhr-Universität Bochum (1995) http://www.num1.ruhr-uni-bochum.de/rv/preprints.html
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Additional Information:
R.
Verfürth
Affiliation:
Fakultät für Mathematik, Ruhr--Universität Bochum, D-44780 Bochum, Germany
Email:
rv@silly.num1.ruhr-uni-bochum.de
DOI:
10.1090/S0025-5718-98-01011-4
PII:
S 0025-5718(98)01011-4
Keywords:
A posteriori error estimates; quasilinear parabolic pdes; space-time finite elements; $\theta $-scheme
Received by editor(s):
March 21, 1995
Received by editor(s) in revised form:
May 3, 1996 and January 3, 1997
Copyright of article:
Copyright
1998,
American Mathematical Society
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