  In this paper, we develop a multigrid preconditioner to solve Darcy flow in
highly heterogeneous porous media. The key component of the preconditioner is
to construct a sequence of nested subspaces $W_{\mathcal{L}}\subset
W_{\mathcal{L}-1}\subset\cdots\subset W_1=W_h$. An appropriate spectral problem
is defined in the space of $W_{i-1}$, then the eigenfunctions of the spectral
problems are utilized to form $W_i$. The preconditioner is applied to solve a
positive semidefinite linear system which results from discretizing the Darcy
flow equation with the lowest order Raviart-Thomas spaces and adopting a
trapezoidal quadrature rule. Theoretical analysis and numerical investigations
of this preconditioner will be presented. In particular, we will consider
several typical highly heterogeneous permeability fields whose resolutions are
up to $1024^3$ and examine the computational performance of the preconditioner
in several aspects, such as strong scalability, weak scalability, and
robustness against the contrast of the media. We also demonstrate an
application of this preconditioner for solving a two-phase flow benchmark
problem.
