Multiscale model reduction and two-level Schwarz preconditioner for H(curl) elliptic problems

TL;DR

This paper introduces a multiscale spectral finite element method combined with a two-level Schwarz preconditioner to efficiently solve $H( ext{curl})$ elliptic problems with heterogeneous coefficients, achieving exponential convergence and effective dimension reduction.

Contribution

It develops a broad-application multiscale spectral GFEM and formulates it as a preconditioner for iterative solvers, demonstrating exponential convergence and superior performance in complex geometries.

Findings

Exponential convergence of the MS-GFEM with respect to local degrees of freedom.

Effective two-level Schwarz preconditioner improves GMRES convergence.

Numerical results show significant dimensionality reduction in 2D and 3D cases.

Abstract

This paper addresses the efficient solution of linear systems arising from curl-conforming finite element discretizations of $H(\mathrm{curl})$ elliptic problems with heterogeneous coefficients. We first employ the discrete form of a multiscale spectral generalized finite element method (MS-GFEM) for model reduction and prove that the method exhibits exponential convergence with respect to the number of local degrees of freedom. The proposed method and its convergence analysis are applicable in broad settings, including general heterogeneous ($L^{\infty}$) coefficients, domains and subdomains with nontrivial topology, irregular subdomain geometries, and high-order finite element discretizations. Furthermore, we formulate the method as an iterative solver, yielding a two-level restricted additive Schwarz type preconditioner based on the MS-GFEM coarse space. The GMRES algorithm, applied to the preconditioned system, is shown to converge at a rate of at least $\Lambda$, where $\Lambda$ denotes the error bound of the discrete MS-GFEM approximation. Numerical experiments in both two and three dimensions demonstrate the superior performance of the proposed methods in terms of dimensionality reduction.