Analysis of search-extension method for finding multiple solutions of nonlinear problem  

作　　者：CHEN ChuanMiao XIE ZiQing 

机构地区：[1]Institute of Computation,Hunan Normal University,Changsha,410081 China

出　　处：《Science China Mathematics》2008年第1期42-54,共13页中国科学：数学（英文版）

基　　金：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 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 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.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.

关 键 词：nonlinear isolated solution REGULARITY search-extension finite elements error analysis 65N30 

分 类 号：O175[理学—数学]