Unconditional Optimal Error Estimates of Linearized, Decoupled and Conservative Galerkin FEMs for the Klein–Gordon–Schrödinger Equation

This paper is concerned with unconditionally optimal error estimates of linearized leap-frog Galerkin finite element methods (FEMs) to numerically solve the d-dimensional (d= 2 , 3) nonlinear Klein–Gordon–Schrödinger (KGS) equation. The proposed FEMs not only conserve the mass and energy in the given discrete norm but also are efficient in implementation because only two linear systems need to be solved at each time step. Meanwhile, an optimal error estimate for the proposed methods is derived by using the temporal-spatial error splitting techniques, which split the error between the exact solution and the numerical solution into two parts, i.e., the temporal error and the spatial error. Since the spatial error is τ-independent, the boundedness of the numerical solution in L^∞-norm follows an inverse inequality immediately without any restriction on the grid ratios. Then, the optimal L² error estimates for r-order FEMs are derived unconditionally. Numerical results in both two and three dimensional spaces are given to confirm the theoretical predictions and demonstrate the efficiency of the methods.