A linearized conservative nonconforming FEM for nonlinear Klein-Gordon-Schrödinger equations

Abstract

A linearized mass and energy-conservative nonconforming FEM is proposed for the nonlinear Klein-Gordon-Schrödinger (KGS) equations. Firstly, by introducing an auxiliary time-discrete system, the error is separated into two parts, i.e., one from the temporal discretization and the other from the spatial discretization. Then, the uniform bounds of the solution to the above time-discrete system in some strong norms and the error estimates in temporal direction are derived through rigorous analysis. Secondly, superclose and superconvergence error estimates in the sense of broken H1-norm are obtained without any restrictions on the time step-size. There are two ingredients in our analysis. One is that the Lp(2≤p<∞)-norm instead of L∞-norm is utilized to bound numerical solutions. The other is that the special characters of EQ1rot element (Lemma 2) are applied to cope with the consistent errors based on the H2 regularity of the time-discrete system. Particularly, sharp error estimates are also employed to deal with the tough nonlinear terms. Finally, numerical tests are given to illustrate our theoretical results.