We present a multiscale mixed finite element method for solving second order elliptic equations with general $L^{\infty}$-coefficients arising from flow in highly heterogeneous porous media. Our approach is based on a multiscale spectral generalized finite element method (MS-GFEM) and exploits the superior local mass conservation properties of mixed finite elements. Following the MS-GFEM framework, optimal local approximation spaces are built for the velocity field by solving local eigenvalue problems over generalized harmonic spaces. The resulting global velocity space is then enriched suitably to ensure inf-sup stability. We develop the mixed MS-GFEM for both continuous and discrete formulations, with Raviart-Thomas based mixed finite elements underlying the discrete method. Exponential convergence with respect to local degrees of freedom is proven at both the continuous and discrete levels. Numerical results are presented to support the theory and to validate the proposed method.

翻译：本文提出了一种多尺度混合有限元方法，用于求解由高度非均质多孔介质流动产生的具有一般$L^{\infty}$系数的二阶椭圆方程。该方法基于多尺度谱广义有限元法（MS-GFEM），并利用了混合有限元优越的局部质量守恒性质。遵循MS-GFEM框架，通过在广义调和空间上求解局部特征值问题，为速度场构建了最优局部逼近空间。随后对生成的全局速度空间进行适当扩充，以确保inf-sup稳定性。我们针对连续和离散两种公式发展了混合MS-GFEM，其中离散方法基于Raviart-Thomas混合有限元。在连续和离散层面均证明了对局部自由度的指数收敛性。文中给出了数值结果以支持理论分析并验证所提方法的有效性。