| Title:
|
Semigroup formulation of Rothe's method: application to parabolic problems (English) |
| Author:
|
Slodička, Marián |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
33 |
| Issue:
|
2 |
| Year:
|
1992 |
| Pages:
|
245-260 |
| . |
| Category:
|
math |
| . |
| Summary:
|
A semilinear parabolic equation in a Banach space is considered. The purpose of this paper is to show the dependence of an error estimate for Rothe's method on the regularity of initial data. The proofs are done using a semigroup theory and Taylor spectral representation. (English) |
| Keyword:
|
error estimates |
| Keyword:
|
parabolic equation |
| Keyword:
|
backward Euler method |
| MSC:
|
35G10 |
| MSC:
|
35K22 |
| MSC:
|
35K25 |
| MSC:
|
47D03 |
| MSC:
|
65M15 |
| MSC:
|
65M20 |
| idZBL:
|
Zbl 0756.65121 |
| idMR:
|
MR1189655 |
| . |
| Date available:
|
2009-01-08T17:55:21Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118492 |
| . |
| Reference:
|
[1] Choudury G.: Fully discrete Galerkin approximations of parabolic boundary-value problems with nonsmooth boundary data.Numer. Math. 57 (1990), 179-203. Zbl 0671.65092, MR 1048311 |
| Reference:
|
[2] Crouzeix M., Thomée V.: On the discretization in time of semilinear parabolic equations with nonsmooth initial data.Math. Comp. 49 (1987), 359-377. MR 0906176 |
| Reference:
|
[3] Henry D.: Geometric Theory of Semilinear Parabolic Equations.Lecture Notes in Math. 840, Springer Verlag, Berlin-Heidelberg-New York, 1981. Zbl 0663.35001, MR 0610244 |
| Reference:
|
[4] Kačur J.: Method of Rothe in Evolution Equations.Teubner Texte zur Math. 80, Leipzig, 1985. MR 0834176 |
| Reference:
|
[5] Kačur J.: Application of Rothe's method to evolution Integrodifferential equations.J. reine angew. Math. 388 (1988), 73-105. Zbl 0638.65098, MR 0944184 |
| Reference:
|
[6] Kačur J.: On $L_\infty $-convergence of Rothe's method.Comment. Math. Univ. Carolinae 30 (1989), 505-510. MR 1031868 |
| Reference:
|
[7] Le Roux M.N.: Semidiscretization in time for parabolic problems.Math. Comp. 33 (1979), 919-931. Zbl 0417.65049, MR 0528047 |
| Reference:
|
[8] Le Roux M.N., Thomée V.: Numerical solution of semilinear integrodifferential equations of parabolic type with nonsmooth data.SIAM J. Numer. Anal. 26 (1989), 1291-1309. MR 1025089 |
| Reference:
|
[9] Luskin M., Rannacher R.: On the smoothing property of the Galerkin method for parabolic equations.SIAM J. Numer. Anal. 19 (1982), 93-113. Zbl 0483.65064, MR 0646596 |
| Reference:
|
[10] Mingyou H., Thomée V.: On the backward Euler method for parabolic equations with rough initial data.SIAM J. Numer. Anal. 19 (1982), 599-603. MR 0656473 |
| Reference:
|
[11] Rannacher R.: $L_\infty $-Stability Estimates and Asymptotic Error Expansion for Parabolic Finite Element Equations.preprint 589, Universität Heidelberg, Sonderforschungsbereich 123, Stochastische mathematische Modelle, 1990. MR 1185533 |
| Reference:
|
[12] Schatz A.H., Thomée V., Wahlbin L.B.: Maximum norm stability and error estimates in parabolic finite element equations.Comm. Pure Appl. Math. 33 (1980), 265-304. MR 0562737 |
| Reference:
|
[13] Slodička M.: Error estimate of approximate solution for a quasilinear parabolic integrodifferential equation in the $L_p$-space.Apl. Mat. 34 (1989), 439-448. MR 1026508 |
| Reference:
|
[14] Slodička M.: An investigation of convergence and error estimate of approximate solution for a quasilinear parabolic integrodifferential equation.Apl. Mat. 35 (1990), 16-27. MR 1039408 |
| Reference:
|
[15] Slodička M.: Error Estimates for Discretization in Time to Linear Homogeneous Parabolic Equations with Nonsmooth Initial data.preprint JINR E5-90-8, Dubna, 1990. |
| Reference:
|
[16] Slodička M.: Smoothing effect and discretization in time to semilinear parabolic equations with nonsmooth data.Comment. Math. Univ. Carolinae 32 (1991), 703-713. MR 1159817 |
| Reference:
|
[17] Taylor A.E.: Introduction to Functional Analysis.John Wiley & Sons, Inc., New York, 1958. Zbl 0654.46002, MR 0098966 |
| Reference:
|
[18] Thomée V.: Galerkin Finite Element Methods for Parabolic Problems.Lecture Notes in Math. 1054, Springer Verlag, Berlin-Heidelberg-New York-Tokyo, 1984. MR 0744045 |
| Reference:
|
[19] Thomée V.: On the Numerical Solution of Integro-Differential Equations of Parabolic Type.Inter. Ser. Numer. Math. 86, Birkhauser Verlag Basel, 1988, 477-493. MR 1022978 |
| . |
