  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.
