An improved search-extension method for computing multiple solutions of semilinear PDEs

Abstract

Most numerical methods for calculating the saddle-point solutions of semilinear elliptic equations are based on various minimax theorems. In these cases, only those with 'nice properties' can be numerically approximated. This paper is geared to combine the finite element method, the interpolated coefficient finite element method, the eigenfunction expansion method and the search-extension method proposed by Chen & Xie (2004, Comput. Math. Appl., 47, 327-343) to obtain multiple solutions for semilinear elliptic equations. This strategy not only greatly reduces the expensive computation, but also successfully obtains multiple solutions for a class of semilinear elliptic boundary value problems with odd or non-odd non-linearity on some symmetric and non-symmetric domains. Numerical solutions which are illustrated by their graphs for visualization will show the efficiency of our approach.