# A convergence analysis of Generalized Multiscale Finite Element Methods

## TL;DR

This paper provides a comprehensive convergence analysis for the Generalized Multiscale Finite Element Method (GMsFEM), demonstrating its effectiveness for solving second-order elliptic equations with heterogeneous multiscale coefficients.

## Contribution

The paper introduces a novel, general convergence analysis for GMsFEM that simplifies previous methods and extends to more sophisticated variants.

## Key findings

- GMsFEM converges reliably for multiscale elliptic problems.
- Error estimates depend on eigenvalues of unused eigenvectors.
- Numerical experiments confirm theoretical convergence and stability.

## Abstract

In this paper, we consider an approximation method, and a novel general analysis, for second-order elliptic differential equations with heterogeneous multiscale coefficients. We obtain convergence of the Generalized Multi-scale Finite Element Method (GMsFEM) method that uses local eigenvectors in its construction. The analysis presented here can be extended, without great difficulty, to more sophisticated GMsFEMs. For concreteness, the obtained error estimates generalize and simplify the convergence analysis of [J. Comput. Phys. 230 (2011), 937-955]. The GMsFEM method construct basis functions that are obtained by multiplication of (approximation of) local eigenvectors by partition of unity functions. Only important eigenvectors are used in the construction. The error estimates are general and are written in terms of the eigenvalues of the eigenvectors not used in the construction. The error analysis involve local and global norms that measure the decay of the expansion of the solution in terms of local eigenvectors. Numerical experiments are carried out to verify the feasibility of the approach with respect to the convergence and stability properties of the analysis in view of the good scientific computing practice.