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

For numerical computations of multiple solutions of the nonlinear elliptic problem Delta u + f(u) = 0 in Omega, u = 0 on Gamma, 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 is an element of H1+alpha, alpha > 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.