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 \(\tau \)-independent, the boundedness of the numerical solution in \(L^\infty \)-norm follows an inverse inequality immediately without any restriction on the grid ratios. Then, the optimal \(L^2\) 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.

本文涉及线性化跳蛙式Galerkin有限元方法（FEM）的无条件最优误差估计，以数值求解d维\（（d = 2,3）\）非线性Klein-Gordon-Schrödinger（KGS）方程。所提出的有限元方法不仅在给定的离散范数下节省了质量和能量，而且由于每个时间步仅需要求解两个线性系统，因此在实现方面效率很高。同时，利用时空误差分割技术推导了所提方法的最优误差估计，将精确解和数值解之间的误差分为时间误差和空间误差两部分。由于空间误差为\（\ tau \）-独立，\（L ^ \ infty \）-范数中数值解的有界性立即遵循逆不等式，而对网格比率没有任何限制。然后，无条件导出r阶有限元的最优\（L ^ 2 \）误差估计。给出二维和三维空间中的数值结果，以证实理论预测并证明该方法的有效性。