Spectral ACMS: A robust localized Approximated Component Mode Synthesis Method

TL;DR

This paper introduces a robust localized spectral method for multiscale finite element approximation of elliptic PDEs with heterogeneous coefficients, achieving contrast-independent optimal error estimates.

Contribution

It develops a spectral ACMS method that combines corner basis functions with local eigenmodes to ensure robustness and optimal convergence regardless of coefficient contrast.

Findings

Achieves contrast-independent optimal error estimates.

Converges at optimal rate for low-regularity solutions.

Provides a robust multiscale finite element framework.

Abstract

We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous $L^\infty$ coefficients. The methods are of Galerkin type and follow the Variational Multiscale and Localized Orthogonal Decomposition--LOD approaches in the sense that it decouples spaces into \emph{multiscale} and \emph{fine} subspaces. In a first method, the multiscale basis functions are obtained by mapping coarse basis functions, based on corners used on primal iterative substructuring methods, to functions of global minimal energy. This approach delivers quasi-optimal a priori error energy approximation with respect to the mesh size, but it is not robust with respect to high-contrast coefficients. In a second method, edge modes based on local generalized eigenvalue problems are added to the corner modes. As a result, optimal a priori error energy estimate is achieved which is mesh and contrast independent. The methods converge at optimal rate even if the solution has minimum regularity, belonging only to the Sobolev space $H^1$.