Edge Multiscale Methods for elliptic problems with heterogeneous coefficients

TL;DR

This paper introduces two novel edge multiscale methods, ESMsFEM and WEMsFEM, for solving elliptic PDEs with high-contrast heterogeneous coefficients, demonstrating their convergence and effectiveness through numerical tests.

Contribution

The paper develops and analyzes two new edge multiscale methods, ESMsFEM and WEMsFEM, tailored for high-contrast elliptic problems, with proven convergence rates.

Findings

Convergence rates depend on coarse mesh size, spectral basis functions, and wavelet level.

Numerical tests verify the theoretical convergence and effectiveness.

Methods outperform traditional approaches in high-contrast scenarios.

Abstract

In this paper, we proposed two new types of edge multiscale methods motivated by \cite{GL18} to solve Partial Differential Equations (PDEs) with high-contrast heterogeneous coefficients: Edge spectral multiscale Finte Element method (ESMsFEM) and Wavelet-based edge multiscale Finite Element method (WEMsFEM). Their convergence rates for elliptic problems with high-contrast heterogeneous coefficients are demonstrated in terms of the coarse mesh size $H$, the number of spectral basis functions and the level of the wavelet space $\ell$, which are verified by extensive numerical tests.