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

Chen, Zhiming; Feng, Jia

doi:10.1090/S0025-5718-04-01634-5

An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems
HTML articles powered by AMS MathViewer

by Zhiming Chen and Jia Feng;

Math. Comp. 73 (2004), 1167-1193

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

Published electronically: January 23, 2004

PDF | Request permission

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.

References

Eberhard Bänsch, Local mesh refinement in $2$ and $3$ dimensions, Impact Comput. Sci. Engrg. 3 (1991), no. 3, 181–191. MR 1141298, DOI 10.1016/0899-8248(91)90006-G
M. Bieterman and I. Babuška, The finite element method for parabolic equations. I. A posteriori error estimation, Numer. Math. 40 (1982), no. 3, 339–371. MR 695602, DOI 10.1007/BF01396451
M. Bieterman and I. Babuška, The finite element method for parabolic equations. I. A posteriori error estimation, Numer. Math. 40 (1982), no. 3, 339–371. MR 695602, DOI 10.1007/BF01396451
I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), no. 4, 736–754. MR 483395, DOI 10.1137/0715049
Zhiming Chen and Shibin Dai, Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity, SIAM J. Numer. Anal. 38 (2001), no. 6, 1961–1985. MR 1856238, DOI 10.1137/S0036142998349102
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.
Zhiming Chen, Ricardo H. Nochetto, and Alfred Schmidt, A characteristic Galerkin method with adaptive error control for the continuous casting problem, Comput. Methods Appl. Mech. Engrg. 189 (2000), no. 1, 249–276. MR 1779683, DOI 10.1016/S0045-7825(99)00295-9
Zhiming Chen, Ricardo H. Nochetto, and Alfred Schmidt, Error control and adaptivity for a phase relaxation model, M2AN Math. Model. Numer. Anal. 34 (2000), no. 4, 775–797. MR 1784485, DOI 10.1051/m2an:2000103
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 520174
Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9 (1975), no. no. , no. R-2, 77–84 (English, with French summary). MR 400739
Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. MR 1393904, DOI 10.1137/0733054
Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal. 28 (1991), no. 1, 43–77. MR 1083324, DOI 10.1137/0728003
Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. IV. Nonlinear problems, SIAM J. Numer. Anal. 32 (1995), no. 6, 1729–1749. MR 1360457, DOI 10.1137/0732078
Peter 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), no. 1, 149–169. MR 1259970, DOI 10.1137/0731008
Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488. MR 1770058, DOI 10.1137/S0036142999360044
R. H. Nochetto, A. Schmidt, and C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems, Math. Comp. 69 (2000), no. 229, 1–24. MR 1648399, DOI 10.1090/S0025-5718-99-01097-2
Marco Picasso, Adaptive finite elements for a linear parabolic problem, Comput. Methods Appl. Mech. Engrg. 167 (1998), no. 3-4, 223–237. MR 1673951, DOI 10.1016/S0045-7825(98)00121-2
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.
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, Teubner (1996).
R. Verfürth, A posteriori error estimates for nonlinear problems: $L^r(0,T;W^{1,\rho }(\Omega ))$-error estimates for finite element discretizations of parabolic equations, Numer. Methods Partial Differential Equations 14 (1998), no. 4, 487–518. MR 1627578, DOI 10.1002/(SICI)1098-2426(199807)14:4<487::AID-NUM4>3.0.CO;2-G