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Spectral ACMS: A robust localized Approximated Component Mode Synthesis Method

Alexandre L. Madureira, Marcus Sarkis

TL;DR

The paper develops a robust ACMS framework within the Localized Orthogonal Decomposition for two-dimensional elliptic problems with heterogeneous coefficients. It presents two variants: a Low-Contrast ACMS-LOD that achieves quasi-optimal, mesh- and contrast-insensitive error decay via localized projections, and a High-Contrast ACMS-NLSD/LSD that uses edge-based spectral decompositions to retain only a small set of eigenfunctions, ensuring contrast-robust a priori error estimates. A key contribution is the establishment of exponential decay for localized operators and the design of practical localization schemes with well-controlled error, even in the presence of high-contrast channels, complemented by a spectral problem inside each substructure to efficiently capture multiscale behavior. The approach yields accurate, efficient, and robust multiscale approximations with minimal regularity requirements, and numerical experiments illustrate significant improvements in energy error and localization efficiency across varying contrast scenarios.

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 $H^1$.

Spectral ACMS: A robust localized Approximated Component Mode Synthesis Method

TL;DR

The paper develops a robust ACMS framework within the Localized Orthogonal Decomposition for two-dimensional elliptic problems with heterogeneous coefficients. It presents two variants: a Low-Contrast ACMS-LOD that achieves quasi-optimal, mesh- and contrast-insensitive error decay via localized projections, and a High-Contrast ACMS-NLSD/LSD that uses edge-based spectral decompositions to retain only a small set of eigenfunctions, ensuring contrast-robust a priori error estimates. A key contribution is the establishment of exponential decay for localized operators and the design of practical localization schemes with well-controlled error, even in the presence of high-contrast channels, complemented by a spectral problem inside each substructure to efficiently capture multiscale behavior. The approach yields accurate, efficient, and robust multiscale approximations with minimal regularity requirements, and numerical experiments illustrate significant improvements in energy error and localization efficiency across varying contrast scenarios.

Abstract

We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous 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 .

Paper Structure

This paper contains 8 sections, 16 theorems, 110 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Substructuring Formulation
  3. The Low-Contrast Multiscale Case
  4. Decaying Results for Low-Contrast coefficients
  5. The High-Contrast Multiscale Case
  6. Decaying Results for High-Contrast coefficients
  7. Spectral Multiscale Problems inside Substructures
  8. Numerical Experiment Tests

Key Result

Lemma 1

Let $\tau\in{\mathcal{T}_H}$ and $c_{P,L}^\tau$ as in e:lpi. Then, an upper bound for $c_{P,L}^\tau$ is given by

Figures (2)

  • Figure 1: On the left, the distribution of the coefficient for a $8\times 8$ subdomain decomposition. On the right, the log-normal plot of the multiscale basis functions without adaptivity. Note that there is no exponential decay whatsoever.
  • Figure 2: Log-normal plot showing the decay of a multiscale basis functions with adaptivity, for ${\mathscr{H}} =H$.

Theorems & Definitions (38)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Lemma 5
  • proof
  • ...and 28 more