# An analysis of the NLMC upscaling method for high contrast problems

## TL;DR

This paper introduces a multiscale basis function approach based on the NLMC method for efficiently solving high contrast elliptic problems, with proven convergence and demonstrated numerical effectiveness.

## Contribution

It develops a new local basis construction method within the NLMC framework that captures multiscale and non-local effects for high contrast media.

## Key findings

- Basis functions are localizable with decay properties.
- The method converges and achieves desired accuracy with sufficient oversampling.
- Numerical experiments confirm efficiency in high contrast scenarios.

## Abstract

In this paper we propose simple multiscale basis functions with constraint energy minimization to solve elliptic problems with high contrast medium. Our methodology is based on the recently developed non-local multicontinuum method (NLMC). The main ingredient of the method is the construction of suitable local basis functions with the capability of capturing multiscale features and non-local effects. In our method, each coarse block is decomposed into various regions according to the contrast ratio, and we require that the contrast ratio should be relatively small within each region. The basis functions are constructed by solving a local problem defined on the oversampling domains and they have mean value one on the chosen region and zero mean otherwise. Numerical analysis shows that the resulting basis functions can be localizable and have a decay property. The convergence of the multiscale solution is also proved. Finally, some numerical experiments are carried out to illustrate the performances of the proposed method. They show that the proposed method can solve problem with high contrast medium efficiently. In particular, if the oversampling size is large enough, then we can achieve the desired error.