An analysis of the NLMC upscaling method for high contrast problems
For computational scientists solving multiscale elliptic problems with high contrast, this method improves efficiency and accuracy, but is incremental as it extends an existing framework.
The paper proposes multiscale basis functions with constraint energy minimization for elliptic problems with high contrast media, based on the NLMC method. Numerical experiments show efficient solution and desired error with sufficient oversampling.
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.