Iterative Oversampling Technique for Constraint Energy Minimizing Generalized Multiscale Finite Element Method in the Mixed Formulation

TL;DR

This paper introduces an iterative scheme for constructing multiscale basis functions in the CEM-GMsFEM framework for mixed formulations, improving solution approximation with convergence guarantees.

Contribution

It develops an iterative approach to build multiscale basis functions, combining snapshot spaces, spectral decomposition, and Richardson iteration for enhanced accuracy.

Findings

Achieves first-order convergence with respect to coarse mesh size.

Demonstrates effectiveness through numerical experiments.

Provides an efficient iterative scheme for multiscale basis construction.

Abstract

In this paper, we develop an iterative scheme to construct multiscale basis functions within the framework of the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for the mixed formulation. The iterative procedure starts with the construction of an energy minimizing snapshot space that can be used for approximating the solution of the model problem. A spectral decomposition is then performed on the snapshot space to form global multiscale space. Under this setting, each global multiscale basis function can be split into a non-decaying and a decaying parts. The non-decaying part of a global basis is localized and it is fixed during the iteration. Then, one can approximate the decaying part via a modified Richardson scheme with an appropriately defined preconditioner. Using this set of iterative-based multiscale basis functions, first-order convergence with respect to the coarse mesh size can be shown if sufficiently many times of iterations with regularization parameter being in an appropriate range are performed. Numerical results are presented to illustrate the effectiveness and efficiency of the proposed computational multiscale method.