Postprocessing the finite element method for semilinear parabolic problems

SIAM Journal on Numerical Analysis

Volumn 44, Issue 4, 2006, Pages 1681-1702

  Yan, Yubin ^a  

^a UNIVERSITY OF SHEFFIELD   (United Kingdom)

Author keywords

Backward Euler method; Error estimates; Finite element method; Postprocessing; Semilinear parabolic problem; Time derivative

Indexed keywords

APPROXIMATION THEORY; COST EFFECTIVENESS; ERROR ANALYSIS; EULER EQUATIONS; FINITE ELEMENT METHOD; PROBLEM SOLVING;

BACKWARD EULER METHOD; DIFFERENCE QUOTIENTS; SEMILINEAR PARABOLIC PROBLEM; TIME DERIVATIVES;

LINEAR SYSTEMS;

EID: 34250324786     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142903430931     Document Type: Article

References (30)

1. G. AKRIVIS, M. CROUZEIX, AND C. MAKRIDAKIS, Implicit-explicit multistep finite element methods for nonlinear parabolic problems, Math. Comp., 67 (1998), pp. 457-477.
2. G. AKRIVIS, M. CROUZEIX, AND C. MAKRIDAKIS, Implicit-explicit multistep methods for quasi-linear parabolic equations, Numer. Math., 82 (1999), pp. 521-541.
3. I. BABUŠKA, Finite element method for domains with corners, Computing, 6 (1970), pp. 264-273.
4. S. C. BRENNER AND L. R. SCOTT, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.
5. C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, AND T. A. ZANG, Spectral Methods in Fluid Dynamics, Springer Ser. Comput. Phys., Springer-Verlag, New York, 1988.
6. M. CROUZEIX, V. THOMÉE, AND L. B. WAHLBIN, Error estimates for spatially discrete approximations of semilinear parabolic equations with initial data of low regularity, Math. Comp., 53 (1989), pp. 25-41.
7. J. DE FRUTOS, B. GARCÍA-ARCHILLA, AND J. NOVO, A postprocessed Galerkin method with Chebyshev or Legendre polynomials, Numer. Math., 86 (2000), pp. 419-442.
8. J. DE FRUTOS AND J. NOVO, A postprocess based improvement of the spectral element method, Appl. Numer. Math., 33 (2000), pp. 217-223.
9. J. DE FRUTOS AND J. NOVO, Postprocessing the linear finite element method, SIAM J. Numer. Anal., 40 (2002), pp. 805-819.
10. J. DE FRUTOS AND J. NOVO, A spectral element method for the Navier-Stokes equations with improved accuracy, SIAM J. Numer. Anal., 38 (2000), pp. 799-819.
11. C M. ELLIOTT AND S. LARSSON, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation, Math. Comp., 58 (1992), pp. 603-630.
12. C. M. ELLIOTT AND S. LARSSON, A finite element model for the time-dependent Joule heating problem, Math. Comp., 64 (1995), pp. 1433-1453.
13. C. M. ELLIOTT AND A. M. STUART, The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30 (1993), pp. 1622-1663.
14. B. GARCÍA-ARCHILLA AND E. S. TITI, Postprocessing the Galerkin method: The finite-element case, SIAM J. Numer. Anal., 37 (2000), pp. 470-499.
15. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman, Boston, 1985.
16. H. P. HELFRICH, Error estimates for semidiscrete Galerkin type approximations to semilinear evolution equations with nonsmooth initial data, Numer. Math., 51 (1987), pp. 559-569.
17. C. JOHNSON, S. LARSSON, V. THOMÉE, AND L. B. WAHLBIN, Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data, Math. Comp., 49 (1987), pp. 331-357.
18. S. LARSSON, Nonsmooth Data Error Estimates with Applications to the Study of the Long- Time Behavior of Finite Element Solutions of Semilinear Parabolic Problems, Department of Mathematics, Chalmers University of Technology, 1992-36, preprint.
19. S. LARSSON, The long-time behavior of finite-element approximations of solutions to semilinear parabolic problems, SIAM J. Numer. Anal., 26 (1989), pp. 348-365.
20. S. LARSSON AND J.-M. SANZ-SERNA, The behavior of finite element solutions of semilinear parabolic problems near stationary points, SIAM J. Numer. Anal., 31 (1994), pp. 1000-1018.
21. S. LARSSON AND J.-M. SANZ-SERNA, A shadowing result with applications to finite element approximation of reaction-diffusion equations, Math. Comp., 68 (1999), pp. 55-72.
22. A. H. SCHATZ, I. H. SLOAN, AND L. B. WAHLBIN, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal., 33 (1996), pp. 505-521.
23. A. H. SCHATZ, V. THOMÉE, AND L. B. WAHLBIN, Stability, analyticity, and almost best approximation in maximum norm for parabolic finite element equations, Comm. Pure Appl. Math., 51 (1998), pp. 1349-1385.
24. 1-projection into finite element spaces, Math. Comp., 38 (1982), pp. 1-22.
25. R. SCOTT, Interpolated boundary conditions in the finite element method, SIAM J. Numer. Anal., 12 (1975), pp. 404-427.
26. ∞ estimates for the finite element method on irregular meshes, Math. Comp., 30 (1976), pp. 681-697.
27. V. THOMÉE, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin, 1997.
28. V. THOMÉE AND L. WAHLBIN, On Galerkin methods in semilinear parabolic problems, SIAM J. Numer. Anal., 12 (1975), pp. 378-389.
29. 2 error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., 10 (1973), pp. 723-759.
30. Y. YAN, Smoothing properties and approximation of time derivatives for parabolic equations: Constant time steps, IMA J. Numer. Anal., 23 (2003), pp. 465-487.