A multiscale method for inhomogeneous elastic problems with high contrast coefficients

TL;DR

This paper introduces a novel multiscale finite element method for elasticity problems in high contrast media, effectively handling mixed boundary conditions and demonstrating convergence and accuracy through numerical experiments.

Contribution

The paper develops the first convergence proof for CEM-GMsFEM with mixed boundary conditions in elasticity problems and highlights its contrast-independent accuracy.

Findings

Method's accuracy depends on oversampling domain size.

Contrast independence in target region's precision.

Numerical experiments confirm theoretical results.

Abstract

In this paper, we develop the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with mixed boundary conditions (Dirichlet and Neumann) for the elasticity equations in high contrast media. By a special treatment of mixed boundary conditions separately, and combining the construction of the relaxed and constraint version of the CEM-GMsFEM, we discover that the method offers some advantages such as the independence of the target region's contrast from precision, while the sizes of oversampling domains have a significant impact on numerical accuracy. Moreover, to our best knowledge, this is the first proof of the convergence of the CEM-GMsFEM with mixed boundary conditions for the elasticity equations given. Some numerical experiments are provided to demonstrate the method's performance.