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

Lakkis, Omar; Makridakis, Charalambos

doi:10.1090/S0025-5718-06-01858-8

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

Authors: Omar Lakkis and Charalambos Makridakis
Journal: Math. Comp. 75 (2006), 1627-1658
MSC (2000): Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-06-01858-8
Published electronically: May 26, 2006
MathSciNet review: 2240628
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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. Slimane Adjerid, Joseph E. Flaherty, and Ivo Babuška, A posteriori error estimation for the finite element method-of-lines solution of parabolic problems, Math. Models Methods Appl. Sci. 9 (1999), no. 2, 261–286. MR 1674560, https://doi.org/10.1142/S0218202599000142
2. Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308
3. Georgios Akrivis, Charalambos Makridakis, and Ricardo H. Nochetto, A posteriori error estimates for the Crank-Nicolson method for parabolic equations, Math. Comp. 75 (2006), no. 254, 511–531. MR 2196979, https://doi.org/10.1090/S0025-5718-05-01800-4
4. Ivo Babuška, Miloslav Feistauer, and Pavel Šolín, On one approach to a posteriori error estimates for evolution problems solved by the method of lines, Numer. Math. 89 (2001), no. 2, 225–256. MR 1855826, https://doi.org/10.1007/PL00005467
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 (2001), no. 35-36, 4691–4712. MR 1840797, https://doi.org/10.1016/S0045-7825(00)00340-6
6. A. Bergam, C. Bernardi, and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations, Math. Comp. 74 (2005), no. 251, 1117–1138. MR 2136996, https://doi.org/10.1090/S0025-5718-04-01697-7
7. Dietrich Braess, Finite elements, 2nd ed., Cambridge University Press, Cambridge, 2001. Theory, fast solvers, and applications in solid mechanics; Translated from the 1992 German edition by Larry L. Schumaker. MR 1827293
8. Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258
9. Zhiming Chen and Jia Feng, An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems, Math. Comp. 73 (2004), no. 247, 1167–1193. MR 2047083, https://doi.org/10.1090/S0025-5718-04-01634-5
10. Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
11. Javier de Frutos and Julia Novo, A posteriori error estimation with the 𝑝-version of the finite element method for nonlinear parabolic differential equations, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 43, 4893–4904. MR 1932022, https://doi.org/10.1016/S0045-7825(02)00419-X
12. W. Dörfler and M. Rumpf, An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation, Math. Comp. 67 (1998), no. 224, 1361–1382. MR 1489969, https://doi.org/10.1090/S0025-5718-98-00993-4
13. Todd Dupont, Mesh modification for evolution equations, Math. Comp. 39 (1982), no. 159, 85–107. MR 658215, https://doi.org/10.1090/S0025-5718-1982-0658215-0
14. 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, https://doi.org/10.1137/0728003
15. Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. II. Optimal error estimates in 𝐿_{∞}𝐿₂ and 𝐿_{∞}𝐿_{∞}, SIAM J. Numer. Anal. 32 (1995), no. 3, 706–740. MR 1335652, https://doi.org/10.1137/0732033
16. 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, https://doi.org/10.1137/0732078
17. Bosco García-Archilla and Edriss S. Titi, Postprocessing the Galerkin method: the finite-element case, SIAM J. Numer. Anal. 37 (2000), no. 2, 470–499. MR 1740770, https://doi.org/10.1137/S0036142998335893
18. John G. Heywood and Rolf 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 (1986), no. 4, 750–777. MR 849281, https://doi.org/10.1137/0723049
19. John G. Heywood and Rolf 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 (1988), no. 3, 489–512. MR 942204, https://doi.org/10.1137/0725032
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. Omar Lakkis and Ricardo H. Nochetto, A posteriori error analysis for the mean curvature flow of graphs, SIAM J. Numer. Anal. 42 (2005), no. 5, 1875–1898. MR 2139228, https://doi.org/10.1137/S0036142903430207
22. Xiaohai Liao and Ricardo H. Nochetto, Local a posteriori error estimates and adaptive control of pollution effects, Numer. Methods Partial Differential Equations 19 (2003), no. 4, 421–442. MR 1980188, https://doi.org/10.1002/num.10053
23. C. Makridakis and R. H. Nochetto.
A posteriori error analysis of a class of dissipative methods for nonlinear evolution problems.
Preprint, 2002.
24. Charalambos Makridakis and Ricardo H. Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems, SIAM J. Numer. Anal. 41 (2003), no. 4, 1585–1594. MR 2034895, https://doi.org/10.1137/S0036142902406314
25. Ricardo H. Nochetto, Giuseppe Savaré, and Claudio Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math. 53 (2000), no. 5, 525–589. MR 1737503, https://doi.org/10.1002/(SICI)1097-0312(200005)53:5<525::AID-CPA1>3.0.CO;2-M
26. 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, https://doi.org/10.1090/S0025-5718-99-01097-2
27. Marco Picasso, Adaptive finite elements for a linear parabolic problem, Comput. Methods Appl. Mech. Engrg. 167 (1998), no. 3-4, 223–237. MR 1673951, https://doi.org/10.1016/S0045-7825(98)00121-2
28. A. Schmidt and K. G. Siebert, ALBERT—software for scientific computations and applications, Acta Math. Univ. Comenian. (N.S.) 70 (2000), no. 1, 105–122. MR 1865363
29. L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, https://doi.org/10.1090/S0025-5718-1990-1011446-7
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^{𝑟}(0,𝑇;𝐿^{𝜌}(Ω))-error estimates for finite element discretizations of parabolic equations, Math. Comp. 67 (1998), no. 224, 1335–1360. MR 1604371, https://doi.org/10.1090/S0025-5718-98-01011-4
32. R. Verfürth, A posteriori error estimates for nonlinear problems: 𝐿^{𝑟}(0,𝑇;𝑊^{1,𝜌}(Ω))-error estimates for finite element discretizations of parabolic equations, Numer. Methods Partial Differential Equations 14 (1998), no. 4, 487–518. MR 1627578, https://doi.org/10.1002/(SICI)1098-2426(199807)14:4<487::AID-NUM4>3.0.CO;2-G