This paper investigates an efficient exponential integrator generalized multiscale finite element method for solving a class of time-evolving partial differential equations in bounded domains. The proposed method first performs the spatial discretization of the model problem using constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM). This approach consists of two stages. First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. The multiscale basis functions are obtained in the second stage using the auxiliary space by solving local energy minimization problems over the oversampling domains. The basis functions have exponential decay outside the corresponding local oversampling regions. We shall consider the first and second-order explicit exponential Runge-Kutta approach for temporal discretization and to build a fully discrete numerical solution. The exponential integration strategy for the time variable allows us to take full advantage of the CEM-GMsFEM as it enables larger time steps due to its stability properties. We derive the error estimates in the energy norm under the regularity assumption. Finally, we will provide some numerical experiments to sustain the efficiency of the proposed method.

翻译：本文研究了一种高效的指数积分器广义多尺度有限元方法，用于求解有界区域中的一类时间演化偏微分方程。所提方法首先采用约束能量最小化广义多尺度有限元方法（CEM-GMsFEM）对模型问题进行空间离散。该方法包含两个阶段：第一阶段通过求解局部谱问题构建辅助空间，其中小特征值对应的基函数被捕获；第二阶段利用辅助空间，通过在过采样域中求解局部能量最小化问题获得多尺度基函数。这些基函数在相应局部过采样区域外呈指数衰减。我们将采用一阶和二阶显式指数龙格-库塔方法进行时间离散，并构建全离散数值解。时间变量的指数积分策略使得我们能够充分利用CEM-GMsFEM的优势——由于其稳定性特性，该方法允许采用更大的时间步长。我们在正则性假设下推导了能量范数下的误差估计。最后，通过数值实验验证了所提方法的有效性。