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

Mathematics of Computation

Volumn 75, Issue 256, 2006, Pages 1627-1658

  Lakkis, Omar ^a   Makridakis, Charalambos ^b,c  

^a UNIVERSITY OF SUSSEX   (United Kingdom)

^b UNIVERSITY OF CRETE   (Greece)

^c INSTITUTE OF COMPUTER SCIENCE   (Greece)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33750850308     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-06-01858-8     Document Type: Article

References (33)

1. S. Adjerid, J. K. 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. MR1674560 (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. MR1885308 (2003b:65001)
3. G. Akrivis, C. Makridakis, and R. H. Norhetto. A posteriori error estimates for the Crank-Nicolson method for parabolic equations. Math. Comp. 75(254):511-531, 2006. MR2196979
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. MR1855826 (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. MR1840797 (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. MR2136996
7. D. Braess. Finite elements. Cambridge University Press, Cambridge, second edition, 2001.
8. Theory, fast solvers, and applications in solid mechanics, Translated from the 1992 German edition by Larry L. Schumaker. MR1827293 (2001k:65002)
9. S. C. Brenner and L. R. Scott. The mathematical theory of finite element methods. Springer-Verlag, New York, 1994. MR1278258 (95f:65001)
10. 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. MR2047083 (2005e:65131)
11. P. G. Ciarlet. The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam, 1978. Studies in Mathematics and its Applications, Vol. 4. MR0520174 (58:25001)
12. J. de Frutos and J. 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(43):4893-4904, 2002. MR1932022 (2003i:65080)
13. W. Dörfler and M. Rumpf. An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation. Math. Comp., 67(224):1361-1382, 1998. MR1489969 (99b:65141)
14. T. Dupont. Mesh modification for evolution equations. Math. Comp., 39(159):85-107, 1982. MR0658215 (84g:65131)
15. 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. MR1083324 (91m:65274)
16. ∞. SIAM J. Numer. Anal., 32(3):706-740, 1995. MR1335652 (96c:65162)
17. K. Eriksson and C. Johnson. Adaptive finite element methods for parabolic problems. IV. Nonlinear problems. SIAM J. Numer. Anal., 32(6):1729-1749, 1995. MR1360457 (96i:65081)
18. 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. MR1740770 (2001h:65112)
19. 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. MR0849281 (88b:65132)
20. 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. MR0942204 (89k:65114)
21. 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.
22. 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. MR2139228
23. 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. MR1980188 (2004c:65130)
24. C. Makridakis and R. H. Nochetto. A posteriori error analysis of a class of dissipative methods for nonlinear evolution problems. Preprint, 2002.
25. 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. MR2034895 (2004k:65157)
26. 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. MR1737503 (2000k:65142)
27. 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. MR1648399 (2000i:65136)
28. M. Picasso. Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg., 167(3-4):223-237, 1998. MR1673951 (2000b:65188)
29. A. Schmidt and K. G. Siebert. ALBERT - software for scientific computations and applications. Acta Math. Univ. Comenian. (N.S.), 70(1):105-122, 2000. MR1865363
30. L. R. Scott and S. Zhang. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp., 54(190):483-493, 1990. MR1011446 (90j:65021)
31. R. Verfürth. A Review of A Posteriori Error Estimation and Adaptive. Mesh-Refinement Techniques. Wiley-Teubner, Chichester-Stuttgart, 1996.
32. ρ(ω))-error estimates for finite element discretizations of parabolic equations. Math. Comp., 67(224):1335-1360, 1998. MR1604371 (99b:65120)
33. 1,ρ(ω))-error estimates for finite element discretizations of parabolic equations. Numer. Methods Partial Differential Equations, 14(4):487-518, 1998. MR1627578 (99g:65099)