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Multiscale Spectral Generalized Finite Element Methods for Discontinuous Galerkin Schemes

Christian Alber, Lukas Holbach

TL;DR

The paper develops a multiscale spectral generalized finite element method (MS-GFEM) for discontinuous Galerkin discretizations of second-order elliptic PDEs with heterogeneous coefficients. Local problems are augmented by corrections from optimally chosen spectral coarse spaces on oversampling regions, assembled globally via a partition of unity to form the MS-GFEM solution $u^G = u^p + u^s$. A central result is nearly exponential decay of the approximation error, expressed through eigenvalue decay $\sqrt{\lambda_{j,n}} \le C_j e^{-c_j n^{1/d}}$ and a Céa-type a priori error bound for $\|u^e-u^G\|_{B^+,\Omega}$, establishing robust multiscale DG accuracy. The framework extends MS-GFEM theory to DG discretizations, enabling stable and accurate multiscale simulations in heterogeneous media.

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 discretized with a weighted symmetric interior-penalty DG scheme.

Multiscale Spectral Generalized Finite Element Methods for Discontinuous Galerkin Schemes

TL;DR

The paper develops a multiscale spectral generalized finite element method (MS-GFEM) for discontinuous Galerkin discretizations of second-order elliptic PDEs with heterogeneous coefficients. Local problems are augmented by corrections from optimally chosen spectral coarse spaces on oversampling regions, assembled globally via a partition of unity to form the MS-GFEM solution . A central result is nearly exponential decay of the approximation error, expressed through eigenvalue decay and a Céa-type a priori error bound for , establishing robust multiscale DG accuracy. The framework extends MS-GFEM theory to DG discretizations, enabling stable and accurate multiscale simulations in heterogeneous media.

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 discretized with a weighted symmetric interior-penalty DG scheme.
Paper Structure (3 sections, 6 theorems, 29 equations)

This paper contains 3 sections, 6 theorems, 29 equations.

Table of Contents

  1. Motivation and Model Setup
  2. Discontinuous Galerkin Method
  3. MS-GFEM for DG Discretizations of Elliptic PDEs

Key Result

Lemma 1

Let $u\in \mathcal{H}(\omega)$, $\chi\in\mathcal{P}_{1}(\omega, \mathbb{T}_h)$. Then, $\|{\chi u - I_h(\chi u)}\|_{\mathcal{H}_0(\omega)} \lesssim \|{\chi u}\|_{\mathcal{H}(\omega)}$.

Theorems & Definitions (13)

  • proof : Verification of Ma2025
  • Lemma 1
  • proof
  • Corollary 2
  • proof : Verification of Ma2025
  • proof : Verification of Ma2025
  • Lemma 3: Caccioppoli inequality, Ma2025
  • proof
  • Lemma 4: Weak approximation property, Ma2025
  • proof
  • ...and 3 more