Convergence analysis of a minimax method for finding multiple solutions of semilinear elliptic equation.: I: On polyhedral domain. (English) Zbl 1320.65156
The authors “point out what approximation problem is, when Li-Zhou’s methods are used to solve semilinear elliptic equation”. The convergence of the solution of the approximation problem is proved.
Reviewer: Răzvan Răducanu (Iaşi)
MSC:
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J61 | Semilinear elliptic equations |
Keywords:
multiple solutions; polyhedral domain; minimax method; semilinear elliptic equation; finite element method; convergenceReferences:
| [1] | Adams, Robert Alexander, Fournier, John J.F.: Sobolev Spaces. Academic Press, New York (2003) · Zbl 1098.46001 |
| [2] | Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349-381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7 |
| [3] | Brezis, H., Nirenberg, L.: Remarks on finding critical points. Commun. Pure Appl. Math. 44, 939-963 (1991) · Zbl 0751.58006 · doi:10.1002/cpa.3160440808 |
| [4] | Chen, X., Lin, T.C., Wei, J.C.: Blowup and solitary wave solutions with ring profiles of two-component nonlinear Schrodinger systems. Physica D 239, 613-626 (2010) · Zbl 1186.37087 · doi:10.1016/j.physd.2010.01.017 |
| [5] | Chen, X., Zhou, J.: A local min-max-orthogonal method for finding multiple solutions to noncooperative elliptic systems. Math. Comput. 79, 2213-2236 (2010) · Zbl 1200.35094 · doi:10.1090/S0025-5718-10-02336-7 |
| [6] | Chen, X., Zhou, J., Yao, X.: A numerical method for finding multiple co-existing solutions to nonlinear cooperative systems. Appl. Numer. Math. 58, 1614-1627 (2008) · Zbl 1158.65081 · doi:10.1016/j.apnum.2007.09.007 |
| [7] | Choi, Y.S., McKenna, P.J.: A mountain pass method for the numerical solution of semilinear elliptic problems. Nonlinear Anal. 20, 417-437 (1993) · Zbl 0779.35032 · doi:10.1016/0362-546X(93)90147-K |
| [8] | Ciarlet, P.G., Lions, J.L.: Handbook of Numerical Analysis. Elsevier Science Publishers B. V, North-Holland (1975) · Zbl 1156.65055 |
| [9] | Ding, Z., Costa, D., Chen, G.: A high linking method for sign changing solutions for semilinear elliptic equations. Nonlinear Anal. 38, 151-172 (1999) · Zbl 0941.35023 · doi:10.1016/S0362-546X(98)00086-8 |
| [10] | Le, A., Wang, Z.Q., Zhou, J.: Finding multiple solutions to elliptic PDE with nonlinear boundary conditions. J. Sci. Comput. 56, 591-615 (2013) · Zbl 1278.65188 · doi:10.1007/s10915-013-9689-9 |
| [11] | Li, Y., Zhou, J.: A minimax method for finding multiple critical points and its applications to nonlinear PDEs. SIAM J. Sci. Comput. 23, 840-865 (2001) · Zbl 1002.35004 · doi:10.1137/S1064827599365641 |
| [12] | Li, Y., Zhou, J.: Convergence results of a local minimax method for finding multiple critical points. SIAM J. Sci. Comput. 24, 865-885 (2002) · Zbl 1040.58003 · doi:10.1137/S1064827500379732 |
| [13] | Ni, W.M.: Some aspects of semilinear elliptic equations. Dept. of Math. National Tsing Hua Univ., Hsinchu, Taiwan, Rep. of China (1987) |
| [14] | Rabinowitz, P.: Minimax Method in Critical Point Theory with Application to Differential Equations, CBMS Regional Conference Series in Mathematics, No. 65, AMS, Providence (1986) · Zbl 0609.58002 |
| [15] | Struwe, M.: Variational Methods. Springer, New York (1996) · Zbl 0864.49001 · doi:10.1007/978-3-662-03212-1 |
| [16] | Wang, C., Zhou, J.: An orthogonal subspace minimization method for finding multiple solutions to defocusing Schrodinger equation with symmetries. NMPDE 29, 1778-1800 (2013) · Zbl 1274.65305 |
| [17] | Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996) · Zbl 0856.49001 · doi:10.1007/978-1-4612-4146-1 |
| [18] | Xie, Z.Q., Yuan, Y.J., Zhou, J.: On finding multiple solutions to a singularly perturbed Neumann problem. SIAM J. Sci. Comput. 34, 395-420 (2012) · Zbl 1238.58008 · doi:10.1137/100810411 |
| [19] | Yao, X.: A minimax method for finding saddle critical points of upper semi-differentiable locally Lipschitz continuous functional in Banach space and its convergence. Math. Comput. 82, 2087-2136 (2013) · Zbl 1273.65092 · doi:10.1090/S0025-5718-2013-02669-5 |
| [20] | Yao, X., Zhou, J.: A minimax method for finding multiple critical points in Banach spaces and its application to quasi-linear elliptic PDE. SIAM J. Sci. Comput. 26, 1796-1809 (2005) · Zbl 1078.58009 · doi:10.1137/S1064827503430503 |
| [21] | Yao, X., Zhou, J.: Unified convergence results on a minimax algorithm for finding multiple critical points in Banach spaces. SIAM J. Num. Anal. 45, 1330-1347 (2007) · Zbl 1143.58006 · doi:10.1137/050627320 |
| [22] | Yao, X., Zhou, J.: Numerical methods for computing nonlinear eigenpairs: Part I. Isohomogeneous cases. SIAM J. Sci. Comput. 29, 1355-1374 (2007) · Zbl 1156.65055 · doi:10.1137/060651859 |
| [23] | Yao, X., Zhou, J.: Numerical methods for computing nonlinear eigenpairs: Part II. Non-Isohomogeneous cases. SIAM J. Sci. Comput. 30, 937-956 (2008) · Zbl 1183.65140 · doi:10.1137/060656425 |
| [24] | Yao, X., Zhou, J.: A local minimax characterization for computing multiple nonsmooth saddle critical points, Ser. B. Math. Program. 104(2-3), 749-760 (2005) · Zbl 1124.90047 |
| [25] | Zhou, J.: Global sequence convergence of a local minimax method for finding multiple solutions in Banach spaces. Num. Funct. Anal. Optim. 32, 1365-1380 (2011) · Zbl 1235.65137 · doi:10.1080/01630563.2011.597630 |
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