SUPERCONVERGENCE OF FEMS AND NUMERICAL CONTINUATION FOR PARAMETER-DEPENDENT PROBLEMS WITH FOLDS

Abstract

We study finite element approximations for positive solutions of semilinear elliptic eigenvalue problems with folds, and exploit the superconvergence of finite element methods (FEM). In order to apply the superconvergence of FEM for Poisson's equation in [Chen & Huang, 1995; Huang et al., 2004, 2006; Lin & Yan, 1996] to parameter-dependent problems with folds, this paper provides the framework of analysis, accompanied with the proof of the strong monotonicity of the nonlinear form. It is worthy to point out that the superconvergence of the nonlinear problem in this paper is different from that in [Chen & Huang, 1995]. A continuation algorithm is described to trace solution curves of semilinear elliptic eigenvalue problems, where the Adini elements are exploited to discretize the PDEs. Numerical results on some sample test problems with folds and bifurcations are reported.