Superconvergence results for nonlinear Klein-Gordon-Schrodinger equation with backward differential formula finite element method

The main aim of this paper is to derive superconvergence results for nonlinear Klein-Gordon-Schrodinger equation (KGSE) with backward differential formula (BDF) finite element method (FEM). A linearized fully discretized scheme is presented to approximate the solution of the nonlinear equations. To get rid of the restriction about the ratio between h andr, a time-discrete system is recommended to split the error into temporal error and spatial error. Based on the detailed investigation, the technique of recombination for some terms gives the chance to bound the temporal errors in H-2-norm and the spatial errors in H-1-norm, respectively. By virtue of the Ritz projection and the interpolation operator together, superconvergence results for spatial error of order K(h(2) + r(2)) in H-1-norm for the original variable are deduced based on the spatial errors. Finally, numerical example is provided to support the theoretical analysis. Here, h is the subdivision parameter, and ris the time step.