In this paper, numerical solutions for linear Riesz space fractional partial differential equations with a second-order time derivative are considered. A space-time finite element method is proposed to solve these equations numerically. In the time direction, the C⁰-continuous Galerkin method is used to approximate the second-order time derivative. In the space direction, the usual linear finite element method is developed to approximate the space fractional derivative. The matrix equivalent form of this numerical method is deduced. The stability of the discrete solution is established and the optimal error estimates are investigated. Some numerical tests are given to validate the theoretical results.

本文考虑了具有二阶时间导数的线性Riesz空间分数阶偏微分方程的数值解。提出了一种时空有限元方法来数值求解这些方程。在时间方向上，使用C ^0-连续Galerkin方法近似二阶时间导数。在空间方向上，开发了通常的线性有限元方法来近似空间分数导数。推导了该数值方法的矩阵等效形式。建立了离散解的稳定性，并研究了最佳误差估计。进行了一些数值测试，以验证理论结果。