In this paper, for solving a class of linear parabolic equations in rectangular domains, we have proposed an efficient Parareal exponential integrator finite element method. The proposed method first uses the finite element approximation with continuous multilinear rectangular basis function for spatial discretization, and then takes the Runge-Kutta approach accompanied with Parareal framework for time integration of the resulting semi-discrete system to produce parallel-in-time numerical solution. Under certain regularity assumptions, fully-discrete error estimates in $L^2$-norm are derived for the proposed schemes with random interpolation nodes. Moreover, a fast solver can be provided based on tensor product spectral decomposition and fast Fourier transform (FFT), since the mass and coefficient matrices of the proposed method can be simultaneously diagonalized with an orthogonal matrix. A series of numerical experiments in various dimensions are also presented to validate the theoretical results and demonstrate the excellent performance of the proposed method.

翻译：本文针对矩形区域上一类线性抛物型方程的求解，提出了一种高效的Parareal指数积分有限元方法。该方法首先采用连续多线性矩形基函数的有限元逼近进行空间离散，随后结合Runge-Kutta方法与Parareal框架对所得半离散系统进行时间积分，以生成时间并行数值解。在一定的正则性假设下，推导了采用随机插值节点时该格式在$L^2$范数下的全离散误差估计。此外，由于所提方法的质量矩阵与系数矩阵可通过正交矩阵同时对角化，基于张量积谱分解与快速傅里叶变换（FFT）可构造快速求解器。文中还展示了不同维度的系列数值实验，验证了理论结果并证明了所提方法的优异性能。