In this paper, we propose a linearized finite element method for solving two-dimensional fractional Klein-Gordon equations with a cubic nonlinear term. The employed time discretization is a weighted combination of the L2 − 1_σ formula introduced recently by Lyu and Vong (Numer. Algorithms 78(2):485–511, 2018), Galerkin finite element method is used for the spatial discretization, and the cubic nonlinear term is handled explicitly. Using mathematical induction, we prove that the numerical solution is bounded and the fully discrete scheme is convergent with second-order accuracy in time. In numerical experiments, some problems with both smooth and non-smooth exact solutions are considered.

在本文中，我们提出了一种线性有限元方法，用于求解带有立方非线性项的二维分数阶Klein-Gordon方程。所采用的时间离散是的加权组合大号2 - 1点_σ式由吕和疯（NUMER算法。最近推出78（2）：485-511，2018），Galerkin有限元方法被用于空间离散，和三次非线性项被明确处理。使用数学归纳法，我们证明了数值解是有界的，并且完全离散的方案在时间上收敛于二阶精度。在数值实验中，考虑了光滑和非光滑精确解的一些问题。