Unconditional superconvergence analysis of the conservative linearized Galerkin FEMs for nonlinear Klein-Gordon-Schrödinger equation

Abstract In this paper, we propose the conservative linearized Galerkin finite element methods (FEMs) for the nonlinear Klein-Gordon-Schrodinger equation (KGSE) with homogeneous boundary conditions. The constructed schemes conserve not only the discrete mass but also the discrete energy. By splitting the error into the temporal error and the spatial error, the unconditional superconvergence results are deduced. On the one hand, based on the detailed investigation of the temporal errors skillfully, the regularity of the time-discrete systems is presented. On the other hand, the spatial errors with the scale of O ( h 2 ) in L 2 − norm are derived by applying the classical Ritz projection, which are τ−independent with the above achievements. Then, the superclose estimates with order O ( h 2 + τ 2 ) in the sense of H 1 − norm are arrived at by virtue of the relationship between the Ritz projection and the interpolation. Moreover, the global superconvergence properties are obtained through the interpolated postprocessing technique. Finally, numerical examples are given to test the theoretical results. Here, h and τ denote the subdivision parameter and the time step, respectively.

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