# A two-level overlapping Schwarz method with energy-minimizing multiscale coarse basis functions

## TL;DR

This paper introduces a two-level overlapping Schwarz method utilizing energy-minimizing multiscale basis functions to create a robust preconditioner for elliptic problems with high contrast coefficients, improving efficiency and stability.

## Contribution

It develops a novel multiscale coarse basis construction using local spectral problems and energy minimization, enhancing robustness against high contrast in coefficients.

## Key findings

- Preconditioner is robust to coefficient contrast.
- Local basis functions can be computed efficiently.
- Numerical results confirm theoretical robustness.

## Abstract

A two-level overlapping Schwarz method is developed for second order elliptic problems with highly oscillatory and high contrast coefficients, for which it is known that the standard coarse problem fails to give a robust preconditioner. In this paper, we develop energy minimizing multiscale finite element functions to form a more robust coarse problem. First, a local spectral problem is solved in each non-overlapping coarse subdomain, and dominant eigenfunctions are selected as auxiliary functions, which are crucial for the high contrast case. The required multiscale basis functions are then obtained by minimizing an energy subject to some orthogonality conditions with respect to the auxiliary functions. Due to an exponential decay property, the minimization problem is solved locally on oversampling subdomains, that are unions of a few coarse subdomains. The coarse basis functions are therefore local and can be computed efficiently. The resulting preconditioner is shown to be robust with respect to the contrast in the coefficients as well as the overlapping width in the subdomain partition. Numerical results are presented to validate the theory and show the performance.