A systematic coarse-scale model reduction technique for parameter-dependent flows in highly heterogeneous media and its applications

In this paper, we propose a multiscale approach for solving the parameter-dependent elliptic equation with highly heterogeneous coefficients. In particular, we assume that the coefficients have both small scales and high contrast (where the high contrast refers to the large variations in the coefficients). The main idea of our approach is to construct local basis functions that encode the local features present in the coefficient to approximate the solution of a parameter-dependent flow equation. Constructing local basis functions involves (1) finding initial multiscale basis functions, and (2) constructing local spectral problems for complementing the initial coarse space. We use the reduced basis (RB) approach to construct a reduced dimensional local approximation that allows quickly computing the local spectral problem. This is done following the RB concept by constructing a low dimensional approximation offline. For any online parameter value, we use a reduced dimensional approximation of the local problem to construct multiscale basis functions. These local computations are fast and are used to solve the coarse-scale dimensional problem. We present the details of the algorithm and numerical results. The locally supported basis functions can be used to obtain a coarse multiscale approximation for any smooth right-hand side (source term). The approximation of the solution is obtained by solving a coarse global problem. The coarse problem can also be used to construct robust iterative methods of domain decomposition type. Our numerical results show that one can achieve a substantial dimension reduction when solving the local spectral problems. We discuss convergence of the method, the construction of initial multiscale basis functions, and the computational cost of the proposed method.