Constraint energy minimizing generalized multiscale finite element method for inhomogeneous boundary value problems with high contrast coefficients

TL;DR

This paper introduces the CEM-GMsFEM, a multiscale finite element method designed to efficiently solve elliptic PDEs with inhomogeneous boundary conditions and high contrast coefficients, ensuring contrast-independent convergence.

Contribution

The paper develops a novel multiscale finite element framework with specialized basis functions and operators that handle inhomogeneous boundary conditions and high contrast coefficients effectively.

Findings

Convergence of boundary operators is contrast-independent.

Oversampling layers are crucial for controlling numerical errors.

Experiments confirm method reliability with high contrast ratios.

Abstract

In this article we develop the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for elliptic partial differential equations with inhomogeneous Dirichlet, Neumann, and Robin boundary conditions, and the high contrast property emerges from the coefficients of elliptic operators and Robin boundary conditions. By careful construction of multiscale bases of the CEM-GMsFEM, we introduce two operators $\mathcal{D}^m$ and $\mathcal{N}^m$ which are used to handle inhomogeneous Dirichlet and Neumann boundary values and are also proved to converge independently of contrast ratios as enlarging oversampling regions. We provide a priori error estimate and show that oversampling layers are the key factor in controlling numerical errors. A series of experiments are conducted, and those results reflect the reliability of our methods even with high contrast ratios.