A resourceful splitting technique with applications to deterministic and stochastic multiscale finite element methods

TL;DR

This paper introduces a novel multiscale finite element method using a splitting technique and Green's kernel, effectively reducing computational complexity for deterministic and stochastic elliptic equations.

Contribution

It develops new multiscale basis functions via a splitting-based Green's kernel, enhancing efficiency in solving both deterministic and stochastic elliptic problems.

Findings

Significant reduction in the dimension of the stochastic parameter space.

Effective combination with sparse grid collocation methods.

Confirmed convergence and computational efficiency through numerical experiments.

Abstract

In this paper we use a splitting technique to develop new multiscale basis functions for the multiscale finite element method (MsFEM). The multiscale basis functions are iteratively generated using a Green's kernel. The Green's kernel is based on the first differential operator of the splitting. The proposed MsFEM is applied to deterministic elliptic equations and stochastic elliptic equations, and we show that the proposed MsFEM can considerably reduce the dimension of the random parameter space for stochastic problems. By combining the method with sparse grid collocation methods, the need for a prohibitive number of deterministic solves is alleviated. We rigorously analyze the convergence of the proposed method for both the deterministic and stochastic elliptic equations. Computational complexity discussions are also offered to supplement the convergence analysis. A number of numerical results are presented to confirm the theoretical findings.