Multiscale Spectral Generalized Finite Element Methods for Discontinuous Galerkin Schemes

TL;DR

This paper introduces a multiscale spectral generalized finite element method for discontinuous Galerkin schemes, achieving efficient local approximations and nearly exponential error decay for complex elliptic problems.

Contribution

It develops a novel multiscale spectral approach combining local source solutions with spectral coarse spaces for DG discretizations.

Findings

Nearly exponential decay of approximation error proven

Method effectively handles highly heterogeneous diffusion

Local approximations improve computational efficiency

Abstract

We propose a multiscale spectral generalized finite element method (MS-GFEM) for discontinuous Galerkin (DG) discretizations. The method builds local approximations on overlapping subdomains as the sum of a local source solution and a correction from an optimal spectral coarse space, which is obtained from a generalized eigenproblem. The global solution is then assembled via a partition of unity. We prove nearly exponential decay of the approximation error for second-order elliptic problems with highly heterogeneous diffusion discretized by a weighted symmetric interior-penalty DG scheme.