| Title:
|
Galerkin approximations for the linear parabolic equation with the third boundary condition (English) |
| Author:
|
Faragó, István |
| Author:
|
Korotov, Sergey |
| Author:
|
Neittaanmäki, Pekka |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
48 |
| Issue:
|
2 |
| Year:
|
2003 |
| Pages:
|
111-128 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We solve a linear parabolic equation in $\mathbb{R}^d$, $d \ge 1,$ with the third nonhomogeneous boundary condition using the finite element method for discretization in space, and the $\theta $-method for discretization in time. The convergence of both, the semidiscrete approximations and the fully discretized ones, is analysed. The proofs are based on a generalization of the idea of the elliptic projection. The rate of convergence is derived also for variable time step-sizes. (English) |
| Keyword:
|
linear parabolic equation |
| Keyword:
|
third boundary condition |
| Keyword:
|
finite element method |
| Keyword:
|
semidiscretization |
| Keyword:
|
fully discretized scheme |
| Keyword:
|
elliptic projection |
| MSC:
|
65M12 |
| MSC:
|
65M15 |
| MSC:
|
65M60 |
| idZBL:
|
Zbl 1099.65086 |
| idMR:
|
MR1966344 |
| DOI:
|
10.1023/A:1026042110602 |
| . |
| Date available:
|
2009-09-22T18:12:52Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134522 |
| . |
| Reference:
|
[1] G. Akrivis, V. Dougalis: On a conservative, high-order accurate finite element scheme for the “parabolic” equation.Comput. Acoustics 1 (1989), 17–26. MR 1095058 |
| Reference:
|
[2] R. Bellman: Introduction to Matrix Analysis.SIAM, Philadelphia, 1997. Zbl 0872.15003, MR 1455129 |
| Reference:
|
[3] S. C. Brenner, L. R. Scott: The Mathematical Theory of Finite Element Methods.Springer-Verlag, New York, Inc. A, 1994. MR 1278258 |
| Reference:
|
[4] J. Douglas, T. Dupont: Galerkin methods for parabolic equations.SIAM J. Numer. Anal. 7 (1970), 575–626. MR 0277126, 10.1137/0707048 |
| Reference:
|
[5] J. Douglas, T. Dupont: The numerical solution of waterflooding problems in petroleum engineering by variational methods.Studies in Numerical Analysis 2, SIAM, Philadelphia, 1970. MR 0269141 |
| Reference:
|
[6] J. Douglas, T. Dupont, H. H. Rachford: The application of variational methods to waterflooding problems.J. Canad. Petroleum Tech. 8 (1969), 79–85. |
| Reference:
|
[7] J. Douglas, T. Dupont, M. F. Wheeler: A quasi-projection analysis of Galerkin methods for parabolic and hyperbolic equations.Math. Comp. 32 (1978), 345–362. MR 0495012, 10.1090/S0025-5718-1978-0495012-2 |
| Reference:
|
[8] I. Faragó, S. Korotov, P. Neittaanmäki: Finite element analysis for the heat conduction equation with the third boundary condition.Annales Univ. Sci. Budapest 41 (1998), 183–195. MR 1691927 |
| Reference:
|
[9] G. Fix, N. Nassif: On finite element approximation in time dependent problems.Numer. Math. 19 (1972), 127–135. MR 0311122, 10.1007/BF01402523 |
| Reference:
|
[10] M. Křížek, P. Neittaanmäki: Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications.Kluwer Academic Publishers, 1996. MR 1431889 |
| Reference:
|
[11] M. Křížek, V. Preiningerová: Calculation of the 3D temperature field of synchronous and of induction machines by the finite element method.Elektrotechn. obzor 80 (1991), 78–84. (Czech) |
| Reference:
|
[12] D. P. Laurie: Optimal parameter choice in the $\theta $-method for parabolic equations.J. Inst. Math. Appl. 19 (1977), 119–125. Zbl 0346.65040, MR 0440926, 10.1093/imamat/19.1.119 |
| Reference:
|
[13] H. S. Price, J. C. Cavendich, R. S. Varga: Numerical methods of higher-order accuracy for diffusion-convection equations.Soc. Petroleum Engrg. J. 8 (1968), 293–303. 10.2118/1877-PA |
| Reference:
|
[14] H. S. Price, R. S. Varga: Error bounds for semi-discrete Galerkin approximations of parabolic problems with applications to petroleum reservoir mechanics.Numerical Solution of Field Problems in Continuum Physics, AMS, Providence, 1970, pp. 74–94. MR 0266452 |
| Reference:
|
[15] A. A. Samarskii: Theory of Difference Schemes.Nauka, Moscow, 1977. (Russian) Zbl 0368.65031, MR 0483271 |
| Reference:
|
[16] J. A. Scott, W. L. Seward: Finite difference method for parabolic problems with nonsmooth initial data.Report of Oxford Univ. Comp. Lab. 86/22 (1987). |
| Reference:
|
[17] G. Strang, G. Fix: An Analysis of the Finite Element Method.Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1973. MR 0443377 |
| Reference:
|
[18] V. Thomée: Galerkin Finite Element Methods for Parabolic Problems.Springer, Berlin, 1997. MR 1479170 |
| Reference:
|
[19] R. S. Varga: Functional Analysis and Approximation Theory in Numerical Analysis.SIAM, Philadelphia, 1971. Zbl 0226.65064, MR 0310504 |
| Reference:
|
[20] M. F. Wheeler: A priori $L_2$ error estimates for Galerkin approximations to parabolic partial differential equations.SIAM J. Numer. Anal. 10 (1973), 723–759. MR 0351124, 10.1137/0710062 |
| . |
