Mathematics of Computation

Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems

Author(s): Omar Lakkis; Charalambos Makridakis.
Journal: Math. Comp. 75 (2006), 1627-1658.
MSC (2000): Primary 65N30
Posted: May 26, 2006
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Abstract: We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of $\operatorname{L}_\infty(0,T;\operatorname{L}_2(\Omega))$ and the higher order spaces, $\operatorname{L}_\infty(0,T;\operatorname{H}^1(\Omega))$ and $\operatorname{H}^1(0,T;\operatorname{L}_2(\Omega))$ , with optimal orders of convergence.

1.: S. Adjerid, J. E. Flaherty, and I. Babuška.
A posteriori error estimation for the finite element method-of-lines solution of parabolic problems.
Math. Models Methods Appl. Sci., 9(2):261-286, 1999. MR 1674560 (2000a:65117)
2.: M. Ainsworth and J. T. Oden.
A posteriori error estimation in finite element analysis.
Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308 (2003b:65001)
3.: G. Akrivis, C. Makridakis, and R. H. Nochetto.
A posteriori error estimates for the Crank-Nicolson method for parabolic equations.
Math. Comp, 75(254):511-531, 2006. MR 2196979
4.: I. Babuška, M. Feistauer, and P. Šolín.
On one approach to a posteriori error estimates for evolution problems solved by the method of lines.
Numer. Math., 89(2):225-256, 2001. MR 1855826 (2002f:65131)
5.: I. Babuška and S. Ohnimus.
A posteriori error estimation for the semidiscrete finite element method of parabolic differential equations.
Comput. Methods Appl. Mech. Engrg., 190(35-36):4691-4712, 2001.MR 1840797 (2002d:65093)
6.: A. Bergam, C. Bernardi, and Z. Mghazli.
A posteriori analysis of the finite element discretization of some parabolic equations.
Math. Comp., 74(1117-1138), 2005. MR 2136996
7.: D. Braess.
Finite elements.
Cambridge University Press, Cambridge, second edition, 2001.
Theory, fast solvers, and applications in solid mechanics, Translated from the 1992 German edition by Larry L. Schumaker.MR 1827293 (2001k:65002)
8.: S. C. Brenner and L. R. Scott.
The mathematical theory of finite element methods.
Springer-Verlag, New York, 1994.MR 1278258 (95f:65001)
9.: Z. Chen and J. Feng.
An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems.
Math. Comp., 73(247):1167-1193 (electronic), 2004. MR 2047083 (2005e:65131)
10.: P. G. Ciarlet.
The finite element method for elliptic problems.
North-Holland Publishing Co., Amsterdam, 1978.
Studies in Mathematics and its Applications, Vol. 4. MR 0520174 (58:25001)
11.: J. de Frutos and J. Novo.
A posteriori error estimation with the -version of the finite element method for nonlinear parabolic differential equations.
Comput. Methods Appl. Mech. Engrg., 191(43):4893-4904, 2002. MR 1932022 (2003i:65080)
12.: W. Dörfler and M. Rumpf.
An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation.
Math. Comp., 67(224):1361-1382, 1998. MR 1489969 (99b:65141)
13.: T. Dupont.
Mesh modification for evolution equations.
Math. Comp., 39(159):85-107, 1982. MR 0658215 (84g:65131)
14.: K. Eriksson and C. Johnson.
Adaptive finite element methods for parabolic problems. I. A linear model problem.
SIAM J. Numer. Anal., 28(1):43-77, 1991. MR 1083324 (91m:65274)
15.: K. Eriksson and C. Johnson.
Adaptive finite element methods for parabolic problems. II. Optimal error estimates in ${L}\sb \infty {L}\sb 2$ and ${L}\sb \infty {L}\sb \infty$ .
SIAM J. Numer. Anal., 32(3):706-740, 1995. MR 1335652 (96c:65162)
16.: K. Eriksson and C. Johnson.
Adaptive finite element methods for parabolic problems. IV. Nonlinear problems.
SIAM J. Numer. Anal., 32(6):1729-1749, 1995. MR 1360457 (96i:65081)
17.: B. García-Archilla and E. S. Titi.
Postprocessing the Galerkin method: the finite-element case.
SIAM J. Numer. Anal., 37(2):470-499 (electronic), 2000. MR 1740770 (2001h:65112)
18.: J. G. Heywood and R. Rannacher.
Finite element approximation of the nonstationary Navier-Stokes problem. II. Stability of solutions and error estimates uniform in time.
SIAM J. Numer. Anal., 23(4):750-777, 1986. MR 0849281 (88b:65132)
19.: J. G. Heywood and R. Rannacher.
Finite element approximation of the nonstationary Navier-Stokes problem. III. Smoothing property and higher order error estimates for spatial discretization.
SIAM J. Numer. Anal., 25(3):489-512, 1988. MR 0942204 (89k:65114)
20.: O. Lakkis and C. Makridakis.
A posteriori error control for parabolic problems: Duality and reconstruction methods.
Preprint, Foundation for Research and Technology, Hellas, Heraklion, Greece, 2003.
21.: O. Lakkis and R. H. Nochetto.
A posteriori error analysis for the mean curvature flow of graphs.
SIAM J. Numer. Anal., 42(5):1875-1898, 2005. MR 2139228
22.: X. Liao and R. H. Nochetto.
Local a posteriori error estimates and adaptive control of pollution effects.
Numer. Methods Partial Differential Equations, 19(4):421-442, 2003. MR 1980188 (2004c:65130)
23.: C. Makridakis and R. H. Nochetto.
A posteriori error analysis of a class of dissipative methods for nonlinear evolution problems.
Preprint, 2002.
24.: C. Makridakis and R. H. Nochetto.
Elliptic reconstruction and a posteriori error estimates for parabolic problems.
SIAM J. Numer. Anal., 41(4):1585-1594 (electronic), 2003.MR 2034895 (2004k:65157)
25.: R. H. Nochetto, G. Savaré, and C. Verdi.
A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations.
Comm. Pure Appl. Math., 53(5):525-589, 2000.MR 1737503 (2000k:65142)
26.: R. H. Nochetto, A. Schmidt, and C. Verdi.
A posteriori error estimation and adaptivity for degenerate parabolic problems.
Math. Comp., 69(229):1-24, 2000. MR 1648399 (2000i:65136)
27.: M. Picasso.
Adaptive finite elements for a linear parabolic problem.
Comput. Methods Appl. Mech. Engrg., 167(3-4):223-237, 1998.MR 1673951 (2000b:65188)
28.: A. Schmidt and K. G. Siebert.
ALBERT--software for scientific computations and applications.
Acta Math. Univ. Comenian. (N.S.), 70(1):105-122, 2000. MR 1865363
29.: L. R. Scott and S. Zhang.
Finite element interpolation of nonsmooth functions satisfying boundary conditions.
Math. Comp., 54(190):483-493, 1990. MR 1011446 (90j:65021)
30.: R. Verfürth.
A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques.
Wiley-Teubner, Chichester-Stuttgart, 1996.
31.: R. Verfürth.
A posteriori error estimates for nonlinear problems. ${L}\sp r(0,{T};{L}\sp \rho(\omega))$ -error estimates for finite element discretizations of parabolic equations.
Math. Comp., 67(224):1335-1360, 1998. MR 1604371 (99b:65120)
32.: R. Verfürth.
A posteriori error estimates for nonlinear problems: ${L}\sp r(0,{T};{W}\sp {1,\rho}(\omega))$ -error estimates for finite element discretizations of parabolic equations.
Numer. Methods Partial Differential Equations, 14(4):487-518, 1998. MR 1627578 (99g:65099)

Omar Lakkis
Affiliation: Department of Mathematics, University of Sussex, Brighton, UK-BN1 9RF, United Kingdom
Email: O.Lakkis@sussex.ac.uk

Charalambos Makridakis
Affiliation: Department of Applied Mathematics, University of Crete, GR-71409 Heraklion, Greece; and Institute for Applied and Computational Mathematics, Foundation for Research and Technology-Hellas, Vasilika Vouton P.O. Box 1527, GR-71110 Heraklion, Greece
Email: makr@tem.uoc.gr

DOI: 10.1090/S0025-5718-06-01858-8
PII: S 0025-5718(06)01858-8
Received by editor(s): December 26, 2003
Received by editor(s) in revised form: May 23, 2005
Posted: May 26, 2006
Additional Notes: The first author was supported by the E.U. RTN {\em Hyke} HPRN-CT-2002-00282 and the EU's \emph{MCWave Marie Curie Fellowship} HPMD-CT-2001-00121 during the preparation of this work at FORTH in Heraklion of Crete, Greece.
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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