| Title:
|
Multilevel correction adaptive finite element method for semilinear elliptic equation (English) |
| Author:
|
Lin, Qun |
| Author:
|
Xie, Hehu |
| Author:
|
Xu, Fei |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
60 |
| Issue:
|
5 |
| Year:
|
2015 |
| Pages:
|
527-550 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A type of adaptive finite element method is presented for semilinear elliptic problems based on multilevel correction scheme. The main idea of the method is to transform the semilinear elliptic equation into a sequence of linearized boundary value problems on the adaptive partitions and some semilinear elliptic problems on very low dimensional finite element spaces. Hence, solving the semilinear elliptic problem can reach almost the same efficiency as the adaptive method for the associated boundary value problem. The convergence and optimal complexity of the new scheme can be derived theoretically and demonstrated numerically. (English) |
| Keyword:
|
semilinear elliptic problem |
| Keyword:
|
multilevel correction |
| Keyword:
|
adaptive finite element method |
| MSC:
|
35J61 |
| MSC:
|
62F35 |
| MSC:
|
65B99 |
| MSC:
|
65N30 |
| idZBL:
|
Zbl 06486924 |
| idMR:
|
MR3396479 |
| DOI:
|
10.1007/s10492-015-0110-x |
| . |
| Date available:
|
2015-09-03T10:41:25Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144390 |
| . |
| Reference:
|
[1] Adams, R. A.: Sobolev Spaces.Pure and Applied Mathematics 65. A Series of Monographs and Textbooks Academic Press, New York (1975). Zbl 0314.46030, MR 0450957 |
| Reference:
|
[2] Babuška, I., Miller, A.: A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator.Comput. Methods Appl. Mech. Eng. 61 (1987), 1-40. Zbl 0593.65064, MR 0880421, 10.1016/0045-7825(87)90114-9 |
| Reference:
|
[3] Babuška, I., Rheinboldt, W. C.: Error estimates for adaptive finite element computations.SIAM J. Numer. Anal. 15 (1978), 736-754. Zbl 0398.65069, MR 0483395, 10.1137/0715049 |
| Reference:
|
[4] Babuška, I., Strouboulis, T.: The Finite Element Method and Its Reliability.Numerical Mathematics and Scientific Computation Clarendon Press, Oxford (2001). MR 1857191 |
| Reference:
|
[5] Babuška, I., Vogelius, M.: Feedback and adaptive finite element solution of one-dimensional boundary value problems.Numer. Math. 44 (1984), 75-102. Zbl 0574.65098, MR 0745088, 10.1007/BF01389757 |
| Reference:
|
[6] Brenner, S. C., Scott, L. R.: The Mathematical Theory of Finite Element Methods.Texts in Applied Mathematics 15 Springer, New York (1994). Zbl 0804.65101, MR 1278258, 10.1007/978-1-4757-4338-8_7 |
| Reference:
|
[7] Cascon, J. M., Kreuzer, C., Nochetto, R. H., Siebert, K. G.: Quasi-optimal convergence rate for an adaptive finite element method.SIAM J. Numer. Anal. 46 (2008), 2524-2550. Zbl 1176.65122, MR 2421046, 10.1137/07069047X |
| Reference:
|
[8] Ciarlet, P. G.: The Finite Element Method for Elliptic Problems.Studies in Mathematics and Its Applications. Vol. 4 North-Holland Publishing Company, Amsterdam (1978). Zbl 0383.65058, MR 0520174 |
| Reference:
|
[9] Dörfler, W.: A convergent adaptive algorithm for Poisson's equation.SIAM J. Numer. Anal. 33 (1996), 1106-1124. Zbl 0854.65090, MR 1393904, 10.1137/0733054 |
| Reference:
|
[10] He, L., Zhou, A.: Convergence and optimal complexity of adaptive finite element methods for elliptic partial differential equations.Int. J. Numer. Anal. Model. 8 (2011), 615-640. MR 2805661 |
| Reference:
|
[11] Holst, M., McCammom, J. A., Yu, Z., Zhou, Y., Zhu, Y.: Adaptive finite element modeling techniques for the Possion-Boltzmann equation.Commun. Comput. Phys. 11 (2012), 179-214. MR 2841952, 10.4208/cicp.081009.130611a |
| Reference:
|
[12] Lin, Q., Xie, H.: A multilevel correction type of adaptive finite element method for Steklov eigenvalue problems.Proc. Internat. Conference `Applications of Mathematics', Prague, 2012. In Honor of the 60th Birthday of M. Křížek Academy of Sciences of the Czech Republic, Institute of Mathematics, Prague (2012), 134-143 J. Brandts et al. Zbl 1313.65298, MR 3204407 |
| Reference:
|
[13] Lin, Q., Xie, H.: A multi-level correction scheme for eigenvalue problems.Math. Comput. 84 (2015), 71-88. Zbl 1307.65159, MR 3266953, 10.1090/S0025-5718-2014-02825-1 |
| Reference:
|
[14] Mekchay, K., Nochetto, R. H.: Convergence of adaptive finite element methods for general second order linear elliptic PDEs.SIAM J. Numer. Anal. 43 (2005), 1803-1827. Zbl 1104.65103, MR 2192319, 10.1137/04060929X |
| Reference:
|
[15] Morin, P., Nochetto, R. H., Siebert, K. G.: Data oscillation and convergence of adaptive FEM.SIAM J. Numer. Anal. 38 (2000), 466-488. Zbl 0970.65113, MR 1770058, 10.1137/S0036142999360044 |
| Reference:
|
[16] Morin, P., Nochetto, R. H., Siebert, K. G.: Convergence of adaptive finite element methods.SIAM Rev. 44 (2002), 631-658. Zbl 1016.65074, MR 1980447, 10.1137/S0036144502409093 |
| Reference:
|
[17] Stevenson, R.: Optimality of a standard adaptive finite element method.Found. Comput. Math. 7 (2007), 245-269. Zbl 1136.65109, MR 2324418, 10.1007/s10208-005-0183-0 |
| Reference:
|
[18] Stevenson, R.: The completion of locally refined simplicial partitions created by bisection.Math. Comput. 77 (2008), 227-241. Zbl 1131.65095, MR 2353951, 10.1090/S0025-5718-07-01959-X |
| Reference:
|
[19] Xie, H.: A multilevel correction type of adaptive finite element method for eigenvalue problems.ArXiv:1201.2308 (2012). MR 3204407 |
| . |
