Proper Orthogonal Decomposition Method for Multiscale Elliptic PDEs with Random Coefficients

Abstract

In this paper, we develop an efficient multiscale reduced basis method to solve elliptic PDEs with multiscale and random coefficients in a multi-query setting. Our method consists of offline and online stages. In the offline stage, a small number of reduced multiscale basis functions are constructed within each coarse grid block using the proper orthogonal decomposition (POD) method. Moreover, local tensor spaces are defined to approximate the solution space of the multiscale random PDEs. In the online stage, a weak formulation is derived and discretized using the Galerkin method to compute the solution. Since the multiscale reduced basis functions can efficiently approximate the high-dimensional solution space, our method is very efficient in solving multiscale elliptic PDEs with random coefficients. Convergence analysis of the proposed method is presented, which shows the dependence of the numerical error on the number of snapshots and the truncation threshold in the POD method. Finally, numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale problems with or without scale separation in the physical space.