A Spectral LOD Method for Multiscale Problems with High Contrast

TL;DR

This paper introduces a spectral localized orthogonal decomposition multiscale finite element method tailored for diffusion problems with high contrast coefficients, achieving accuracy comparable to standard methods on complex domains.

Contribution

It develops a novel multiscale finite element approach based on spectral localized orthogonal decomposition for high contrast diffusion problems, with explicit error estimates.

Findings

Performance similar to standard finite element methods

Explicit error estimates established

Effective on smooth or convex domains

Abstract

We present a multiscale finite element method for a diffusion problem with rough and high contrast coefficients. The construction of the multiscale finite element space is based on the localized orthogonal decomposition methodology and it involves solutions of local finite element eigenvalue problems. We show that the performance of the multiscale finite element method is similar to the performance of standard finite element methods for the homogeneous Dirichlet boundary value problem for the Poisson equation on smooth or convex domains.} Simple explicit error estimates are established under conditions that can be verified from the outputs of the computation.