| Title:
|
A new error estimate for a fully finite element discretization scheme for parabolic equations using Crank-Nicolson method (English) |
| Author:
|
Bradji, Abdallah |
| Author:
|
Fuhrmann, Jürgen |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
113-124 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Finite element methods with piecewise polynomial spaces in space for solving the nonstationary heat equation, as a model for parabolic equations are considered. The discretization in time is performed using the Crank-Nicolson method. A new a priori estimate is proved. Thanks to this new a priori estimate, a new error estimate in the discrete norm of $\mathcal {W}^{1,\infty }(\mathcal {L}^2)$ is proved. An $\mathcal {L}^\infty (\mathcal {H}^1)$-error estimate is also shown. These error estimates are useful since they allow us to get second order time accurate approximations for not only the exact solution of the heat equation but also for its first derivatives (both spatial and temporal). Even the proof presented in this note is in some sense standard but the stated $\mathcal {W}^{1,\infty }(\mathcal {L}^2)$-error estimate seems not to be present in the existing literature of the Crank-Nicolson finite element schemes for parabolic equations. (English) |
| Keyword:
|
parabolic equation |
| Keyword:
|
finite element method |
| Keyword:
|
Crank-Nicolson method |
| Keyword:
|
new error estimate |
| MSC:
|
35K05 |
| MSC:
|
35K15 |
| MSC:
|
35K20 |
| MSC:
|
65M15 |
| MSC:
|
65M60 |
| MSC:
|
65N15 |
| MSC:
|
65N30 |
| idZBL:
|
Zbl 06362246 |
| idMR:
|
MR3238827 |
| DOI:
|
10.21136/MB.2014.143841 |
| . |
| Date available:
|
2014-07-14T08:02:32Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143841 |
| . |
| Reference:
|
[1] Burman, E.: Crank-Nicolson finite element methods using symmetric stabilization with an application to optimal control problems subject to transient advection-diffusion equations.Commun. Math. Sci. 9 (2011), 319-329. Zbl 1279.49019, MR 2836849, 10.4310/CMS.2011.v9.n1.a16 |
| Reference:
|
[2] Chatzipantelidis, P., Lazarov, R. D., Thomée, V.: Some error estimates for the lumped mass finite element method for a parabolic problem.Math. Comput. 81 (2012), 1-20. Zbl 1251.65129, MR 2833485, 10.1090/S0025-5718-2011-02503-2 |
| Reference:
|
[3] Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations.Springer Series in Computational Mathematics 23 Springer, Berlin (1994). Zbl 0803.65088, MR 1299729 |
| Reference:
|
[4] Raviart, P. A., Thomas, J. M.: Introduction to the Numerical Analysis of Partial Differential Equations.Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris French (1983). MR 0773854 |
| Reference:
|
[5] Yu, C., Li, Y.: Biquadratic finite volume element methods based on optimal stress points for parabolic problems.J. Comput. Appl. Math. 236 (2011), 1055-1068. Zbl 1242.65180, MR 2854036, 10.1016/j.cam.2011.07.030 |
| . |
