Acta mathematica scientia, Series B >
THE ACCELERATED SEARCH-EXTENSION METHOD FOR COMPUTING MULTIPLE SOLUTIONS OF SEMILINEAR PDEs
Received date: 2008-12-29
Online published: 2009-07-20
Supported by
This reseach was supported by the National Natural Science Foundation of China (10571053, 10871066, 10811120282), Programme for New Century Excellent Talents in University (NCET-06-0712)
In this paper, we propose an accelerated search-extension method (ASEM)based on the interpolated coefficient finite element method, the search-extension method (SEM) and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems, but also reduces the expensive computation greatly. The numerical results in 1-D and 2-D cases will show the efficiency of our approach.
LIU Ti-Wu , XIE Zi-Qing , CHEN Chuan-Miao . THE ACCELERATED SEARCH-EXTENSION METHOD FOR COMPUTING MULTIPLE SOLUTIONS OF SEMILINEAR PDEs[J]. Acta mathematica scientia, Series B, 2009 , 29(4) : 803 -816 . DOI: 10.1016/S0252-9602(09)60071-0
[1] Chen C M, Xie Z Q. Search-extension algorithm for approximating multiple solutions in nonlinear problem.
Acta Scientiarum Naturalism, Hunan Normal Univ, 2000, 23: 95–96
[2] Chen C M, Xie Z Q. Search extension method for multiple solutions of nonlinear problem. Comp Math
Appl, 2004, 47: 327–343
[3] Chen C M, Xie Z Q. Search Extension Method for Multiple Solutions of Nonlinear Problem. Beijing:
Science Press, 2005
[4] Chen C M, Xie Z Q. Analysis of search-extension method for finding multiple solutions of nonlinear
problem. Science in China, Series A: Mathematics, 2008, 51: 42–54
[5] Chen C M, Larsson S, Zhang N Y. Error estimates of optimal order for finite element methods with
interpolated coefficients for the nonlinear heat equation. IMA J Numer Anal, 1989, 9: 507–524
[6] Choi Y S, McKenna P J. A mountain pass method for the numerical solutions of semilinear elliptic
problems. Nonlinear Anal., 1993, 20: 417–437
[7] Ding Z H, Costa D, Chen G. A high-linking algorithm for sign-changing solutions of semilinear ellipic
equations. Nonlinear Anal., 1999, 38: 151–172
[8] Li Y, Zhou J X. A minimax method for finding multiple critical points and its application to semilinear
PDE. SIAM J Sci Comput, 2001, 23: 840–865
[9] Xie Z Q, Chen C M. Multiple solutions and its Morse indices for one-dimensional nonlinear problem. Acta
Mathematicae Applicatae Sinica, English Series, 2004, 20: 317–322
[10] Xie Z Q, Chen C M. The interpolated coefficient FEM and its application in computing the multiple
solutions of semilinear elliptic problems. International J Numer Anal Model, 2005, 2: 97–106
[11] Xie Z Q, Chen C M, Xu Y. An improved search-extension method for computing multiple solutions of
semilinear PDEs. IMA Journal of Numer Anal, 2005, 25: 549–576
[12] Xu J C. A novel two-grid method for semilinear elliptic equations. SIAM J Sci Comput, 1994, 15: 231–237
[13] Xu J C. Two-grid element discrimination techniques for linear and nonlinear PDEs. SIAM J Numer Anal,
1996, 33: 1759–1777
[14] Zlamal M. A finite element solution of the nonlinear heat equation. RAIRO Model Math Anal Numer,
1980, 14: 203–216
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