×

Modified homotopy perturbation method for solving the Stokes equations. (English) Zbl 1219.76034

Summary: A new approach is proposed for the stationary Stokes equations. Based on the homotopy perturbation method, some iterative algorithms are constructed, and four kinds of perturbation cases are considered respectively. Numerical experiments show that these algorithms are simple and effective.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76M10 Finite element methods applied to problems in fluid mechanics
65N99 Numerical methods for partial differential equations, boundary value problems
35Q35 PDEs in connection with fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Temam, R., Navier-Stokes Equations: Theory and Numerical Analysis (1984), North-Holland: North-Holland Amsterdam · Zbl 0568.35002
[2] Bramble, J. H.; Pasciak, J. E.; Vassilev, A. T., Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34, 1072-1092 (1997) · Zbl 0873.65031
[3] Elman, H. C., Preconditioned for saddle point problems arising in computational fluid dynamics, Appl. Numer. Math., 43, 75-89 (2002) · Zbl 1168.76348
[4] He, Y. N.; Xu, J. C.; Zhou, A. H.; Li, J., Local and parallel finite element algorithms for the Stokes problem, Numer. Math., 109, 415-534 (2008)
[5] Quarteroni, A.; Saleri, F.; Veneziani, A., Factorization methods for the numerical approximation of Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 188, 505-526 (2000) · Zbl 0976.76044
[6] Bank, R.; Welfert, B.; Yserentant, H., A class of iterative methods for solving saddle point problems, Numer. Math., 56, 645-666 (1990) · Zbl 0684.65031
[7] Hu, Q. Y.; Zou, J., Nonlinear inexact Uzawa algorithms foe linear and nonlinear saddle-point problems, SIAM J. Optim., 16, 798-825 (2006) · Zbl 1098.65034
[8] Feng, X. L.; He, Y. N.; Meng, J. X., Application of modified homotopy perturbation method for solving the augmented systems, J. Comput. Appl. Math., 231, 288-301 (2009) · Zbl 1173.65024
[9] Feng, X. L.; Shao, L., On the generalized SOR-like methods for saddle point problems, J. Appl. Math. Inform., 28, 663-677 (2010) · Zbl 1295.65035
[10] He, J. H., Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg., 178, 257-262 (1999) · Zbl 0956.70017
[11] He, J. H., Some asymptotic methods for strongly nonlinear equations, Internat. J. Modern Phys. B, 20, 1141-1199 (2006) · Zbl 1102.34039
[12] He, J. H., An elementary introduction to the homotopy perturbation method, Comput. Math. Appl., 57, 410-412 (2009) · Zbl 1165.65374
[13] Liao, S. J.; Tan, Y., A general approach to obtain series solutions of nonlinear differential equations, Stud. Appl. Math., 119, 297-355 (2007)
[14] Feng, X. L.; Mei, L. Q.; He, G. L., An efficient algorithm for solving Troesch’s problem, Appl. Math. Comput., 189, 500-507 (2007) · Zbl 1122.65373
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.