The multiple solutions for one-dimensional cubic nonlinear problem u"+u^3=0, u(0)=u(π)=0 are computed, on the basis of the eigenpairs of -Φ"k=λkΦk, k=1,2,3.... There exist two nonzero solutions ±uk corresponding to each k, and their Morse index MI(k) for 1≤k≤20 is to be exactly determined. It is shown by the numerical results that MI(k)≥k.
Based on two classes of the orthogonal expansions in a triangle, superconvergence of m-degree triangular finite element solution (for even m) and its average gradient (for odd m) at symmetric points for a second order elliptic problem are studied. There are no other superconvergence points independent of the coefficients of elliptic equation.
Superconvergence structures for rectangular and triangular finite elements are summarized. Two debatable issues in Zhu''s paper [18] are discussed. A superclose polynomial to cubic triangular finite element is constructed by area coordinate.
Chuan-miao Chen (Institute of Computation, Hunan Normal University, Changsha 410081, China)
