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Analysis of search-extension method for finding multiple solutions of nonlinear problem
基金项目:This work was supported by the National Major Basic Research Projects (Grant No. G1999032804),the National Natural Science Foundation of China (Grant No.10471038, 10571053),the Research Fonds for Doctor Programme (Grant No. 20050542006),Programme for New Century Excellent Talent in University (GrantNo. NCET-06-0717)
摘    要:For numerical computations of multiple solutions of the nonlinear elliptic problemΔu f(u)=0 inΩ, u=0 onΓ, a search-extension method (SEM) was proposed and systematically studied by the authors. This paper shall complete its theoretical analysis. It is assumed that the nonlinearity is non-convex and its solution is isolated, under some conditions the corresponding linearized problem has a unique solution. By use of the compactness of the solution family and the contradiction argument, in general conditions, the high order regularity of the solution u∈H~(1 α),α>0 is proved. Assume that some initial value searched by suitably many eigenbases is already fallen into the neighborhood of the isolated solution, then the optimal error estimates of its nonlinear finite element approximation are shown by the duality argument and continuation method.

收稿时间:31 May 2005
修稿时间:27 June 2007

Analysis of search-extension method for finding multiple solutions of nonlinear problem
Authors:Chen ChuanMiao  and Xie ZiQing
Institution:(1) Institute of Computation, Hunan Normal University, Changsha, 410081, China
Abstract:For numerical computations of multiple solutions of the nonlinear elliptic problem Δu + f(u = 0 in Ω, u = 0 on Γ, a search-extension method (SEM) was proposed and systematically studied by the authors. This paper shall complete its theoretical analysis. It is assumed that the nonlinearity is non-convex and its solution is isolated, under some conditions the corresponding linearized problem has a unique solution. By use of the compactness of the solution family and the contradiction argument, in general conditions, the high order regularity of the solution uH 1+α, α > 0 is proved. Assume that some initial value searched by suitably many eigenbases is already fallen into the neighborhood of the isolated solution, then the optimal error estimates of its nonlinear finite element approximation are shown by the duality argument and continuation method. This work was supported by the National Major Basic Research Projects (Grant No. G1999032804), the National Natural Science Foundation of China (Grant No.10471038, 10571053) and the Research Fonds for Doctor Programme (Grant No. 20050542006) and Programme for New Century Excellent Talent in University (Grant No. NCET-06-0717)
Keywords:nonlinear  isolated solution  regularity  search-extension  finite elements  error analysis
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