  In this paper, we consider flow simulation in highly heterogeneous media that
has many practical applications in industry. To enhance mass conservation, we
write the elliptic problem in a mixed formulation and introduce a robust
two-grid preconditioner to seek the solution. We first need to transform the
indefinite saddle problem to a positive definite problem by preprocessing
steps. The preconditioner consists of a local smoother and a coarse
preconditioner. For the coarse preconditioner, we design an adaptive spectral
coarse space motivated by the GMsFEM (Generalized Multiscale Finite Element
Method). We test our preconditioner for both Darcy flow and two phase flow and
transport simulation in highly heterogeneous porous media. Numerical results
show that the proposed preconditioner is highly robust and efficient.
