An Iterative Method Based on Generalized Multiscale Finite Element Methods for Parameter-dependent Dual Continuum Model

Abstract

In this paper, we consider an iterative method based on the generalized multiscale finite element method (GMsFEM) to solve the dual continuum model with random inputs. The main idea of iterative methods is to reformulate the original model into another model with parameter-independent diffusion coefficients and parameter-dependent right-hand sides. Subsequently, a fixed-point iteration is used to compute the solution of the reformulated model. To quantify the statistics of the dual continnum model, we need to solve the coupled system for a large number of samples in the stochastic space. Thus the computation is prohibitively expensive. To exploit the advantages from the GMsFEM, we perform the iteration process in reduced GMsFE space to improve the computation efficiency. The proposed iterative method consisits of two stage, including an offline phase and an online phase. In offline stage, we compute local multiscale basis functions in each coarse grid region based on deterministic multiscale characteristics to construct offline spaces. In online phase, a fixed-point iteration method is employed to compute the solution of the reformulated model in the offline spaces. Additionally, convergence analysis is established under some structure conditions. Finally, we present two numerical tests to show the performance of our proposed method and validate the theoretical convergence results.