An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems

Authors:
Zhiming Chen and Jia Feng

Translated by:

Journal:
Math. Comp. **73** (2004), 1167-1193

MSC (2000):
Primary 65N15, 65N30, 65N50

DOI:
https://doi.org/10.1090/S0025-5718-04-01634-5

Published electronically:
January 23, 2004

MathSciNet review:
2047083

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Abstract | References | Similar Articles | Additional Information

Abstract: An efficient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator. An adaptive algorithm with variable time-step sizes and space meshes is proposed and studied which, at each time step, delays the mesh coarsening until the final iteration of the adaptive procedure, allowing only mesh and time-step size refinements before. It is proved that at each time step the adaptive algorithm is able to reduce the error indicators (and thus the error) below any given tolerance within a finite number of iteration steps. The key ingredient in the analysis is a new coarsening strategy. Numerical results are presented to show the competitive behavior of the proposed adaptive algorithm.

**1.**E. Bänsch,*Local mesh refinement in**and**dimensions,*IMPACT Comput. Sci. Engrg. 3 (1991), 181-191. MR**92h:65150****2.**M. Bieterman and I. Babuska,*The finite element method for parabolic equations: (I) a posteriori estimation,*Numer. Math. 40 (1982), 339-371. MR**85d:65052a****3.**M. Bieterman and I. Babuska,*The finite element method for parabolic equations: (II) a posteriori estimation and adaptive approach,*Numer. Math. 40 (1982), 373-406. MR**85d:65052b****4.**I. Babuska and C. Rheinboldt,*Error estimates for adaptive finite element computations,*SIAM J. Numer. Anal. 15 (1978), 736-754. MR**58:3400****5.**Z. Chen and S. Dai,*Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity,*SIAM J. Numer. Anal. 38 (2001), 1961-1985. MR**2002g:65114****6.**Z. Chen and S. Dai,*On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients,*SIAM J. Sci. Comput. 24 (2002), 443-462.**7.**Z. Chen, R.H. Nochetto and A. Schmidt,*A characteristic Galerkin method with adaptive error control for continuous casting problem,*Comput. Methods Appl. Mech. Engrg. 189 (2000), 249-276. MR**2001f:80003****8.**Z. Chen, R.H. Nochetto and A. Schmidt,*Error control and adaptivity for a phase relaxation model,*Math. Model. Numer. Anal. 34 (2000), 775-797. MR**2001i:65117****9.**P.G. Ciarlet,*The Finite Element Method for Elliptic Problems,*North-Holland, Amsterdam, 1978. MR**58:25001****10.**Ph. Clément,*Approximation by finite element functions using local regularization,*RAIRO Anal. Numer. 9 (1975), 77-84. MR**53:4569****11.**W. Dörfler,*A convergent adaptive algorithm for Possion's equations,*SIAM J. Numer. Anal. 33 (1996), 1106-1124. MR**97e:65139****12.**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****13.**K. Eriksson and C. Johnson,*Adaptive finite element methods for parabolic problems IV: Nonlinear problems,*SIAM J. Numer. Anal. 32 (1995), pp. 1729-1749. MR**96i:65081****14.**P.K. Moore,*A posteriori error estimation with finite element semi- and fully discrete methods for nonlinear parabolic equations in one space dimension,*SIAM J. Numer. Anal. 31 (1994), 149-169. MR**94m:65145****15.**P. Morin, R.H. Nochetto and K.G. Siebert,*Data oscillation and convergence of adaptive FEM,*SIAM J. Numer. Anal. 38 (2000), 466-488. MR**2001g:65157****16.**R.H. Nochetto, A. Schmidt and C. Verdi,*A posteriori error estimation and adaptivity for degenerate parabolic problems,*Math. Comp. 69 (2000), 1-24. MR**2000i:65136****17.**M. Picasso,*Adaptive finite elements for a linear parabolic problem,*Comput. Methods Appl. Mech. Engrg. 167 (1998), 223-237. MR**2000b:65188****18.**A. Schmidt and K.G. Siebert,*ALBERT: An adaptive hierarchical finite element toolbox,*IAM, University of Freiburg, 2000.`http://www.mathematik.uni-freiburg.de/IAM/ Research/projectsdz/albert.`**19.**R. Verfürth,*A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques*, Teubner (1996).**20.**R. Verfürth,*A posteriori error estimates for nonlinear problems:**Finite element discretization of parabolic equations,*Numerical Methods in Partial Differential Equations 14 (1998), 487-518. MR**99g:65099**

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Additional Information

**Zhiming Chen**

Affiliation:
LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, Peoples Republic of China

Email:
zmchen@lsec.cc.ac.cn

**Jia Feng**

Affiliation:
Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, Peoples Republic of China

Email:
fjia@lsec.cc.ac.cn

DOI:
https://doi.org/10.1090/S0025-5718-04-01634-5

Received by editor(s):
September 28, 2001

Received by editor(s) in revised form:
January 12, 2003

Published electronically:
January 23, 2004

Additional Notes:
The first author was supported in part by China NSF under the grant 10025102 and by China MOS under the grant G1999032802

Article copyright:
© Copyright 2004
American Mathematical Society