In this paper, we develop the constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) in mixed formulation applied to parabolic equations with heterogeneous diffusion coefficients. The construction of the method is based on two multiscale spaces: pressure multiscale space and velocity multiscale space. The pressure space is constructed via a set of well-designed local spectral problems, which can be solved independently. Based on the computed pressure multiscale space, we will construct the velocity multiscale space by applying constrained energy minimization. The convergence of the proposed method is proved. In particular, we prove that the convergence of the method depends only on the coarse grid size, and is independent of the heterogeneities and contrast of the diffusion coefficient. Four typical types of permeability fields are exploited in the numerical simulations, and the results indicate that our proposed method works well and gives efficient and accurate numerical solutions.

在本文中，我们开发了混合形式的约束能量最小化广义多尺度有限元方法（CEM-GMsFEM），该方法适用于具有非均质扩散系数的抛物线方程。该方法的构造基于两个多尺度空间：压力多尺度空间和速度多尺度空间。压力空间是通过一组精心设计的局部光谱问题构建的，可以独立解决。基于计算出的压力多尺度空间，我们将通过应用约束能量最小化来构建速度多尺度空间。证明了该方法的收敛性。特别是，我们证明了该方法的收敛性仅取决于粗糙的网格大小，并且与扩散系数的异质性和对比度无关。