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A Super-Localized Generalized Finite Element Method

Philip Freese, Moritz Hauck, Tim Keil, Daniel Peterseim

TL;DR

The paper tackles the challenge of solving elliptic PDEs with arbitrarily rough, multi-scale coefficients by introducing the Super-Localized Generalized Finite Element Method (SL-GFEM), which merges the super-localization of the SLOD with a partition-of-unity approach. Local, problem-adapted spaces are constructed via local solution operators on oversampled patches and reduced to low dimension through Kolmogorov $n$-width spectral problems, then glued globally to form a variational space for the coarse-scale problem. The authors provide rigorous a posteriori error analysis linked to the SLOD framework, present a pessimistic but robust a priori analysis based on the LOD, and demonstrate higher-order extensions that improve localization and convergence. They implement the method in Python using gridlod and validate it on challenging high-contrast channel coefficients, showing superior localization, stability, and convergence relative to existing methods, alongside an open-source reproducible workflow. The work advances numerical homogenization by delivering a simple, scalable, and higher-order-capable multiscale method with strong localization properties suitable for complex media.

Abstract

This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method's basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method's applicability for challenging high-contrast channeled coefficients.

A Super-Localized Generalized Finite Element Method

TL;DR

The paper tackles the challenge of solving elliptic PDEs with arbitrarily rough, multi-scale coefficients by introducing the Super-Localized Generalized Finite Element Method (SL-GFEM), which merges the super-localization of the SLOD with a partition-of-unity approach. Local, problem-adapted spaces are constructed via local solution operators on oversampled patches and reduced to low dimension through Kolmogorov -width spectral problems, then glued globally to form a variational space for the coarse-scale problem. The authors provide rigorous a posteriori error analysis linked to the SLOD framework, present a pessimistic but robust a priori analysis based on the LOD, and demonstrate higher-order extensions that improve localization and convergence. They implement the method in Python using gridlod and validate it on challenging high-contrast channel coefficients, showing superior localization, stability, and convergence relative to existing methods, alongside an open-source reproducible workflow. The work advances numerical homogenization by delivering a simple, scalable, and higher-order-capable multiscale method with strong localization properties suitable for complex media.

Abstract

This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method's basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method's applicability for challenging high-contrast channeled coefficients.
Paper Structure (24 sections, 4 theorems, 115 equations, 6 figures)

This paper contains 24 sections, 4 theorems, 115 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Model problem
  3. Preliminaries
  4. Discontinuous finite element spaces
  5. Conforming companion spaces
  6. Partition of unity
  7. Multi-scale method
  8. Local approximation spaces
  9. Global approximation space
  10. A posteriori error analysis
  11. Higher-order SLOD
  12. A posteriori error bound using SLOD
  13. Local approximation error bound using SLOD
  14. (Pessimistic) a priori error analysis
  15. Higher-order LOD
  16. ...and 9 more sections

Key Result

Theorem 5.3

Let a:riesz be satisfied and let $u$ and $u_{H}^{\ell,n}$ denote the solutions to eq:soleq:PUMSLOD, respectively. Then, there exists a constant $C>0$ independent of $H,\ell,$ and $n$, such that, for any $f \in H^k(\mathcal{T}_H)$, $k \in \mathbb N_0$, with $s \coloneqq \min\{k,p+1\}$ and the notation $H^0(\mathcal{T}_H) \coloneqq L^2(\Omega)$ and $\lvert\cdot\vert_{H^0(\mathcal{T}_H)}\coloneqq

Figures (6)

  • Figure 5.1: Singular values $\sigma_k$ of operator $\Pi_H\lvert_Y$ defined in \ref{['eq:op']} for an interior patch, for different pairs of $\ell,$$p$. The singular values $\sigma_{K-J+1}$ relevant for \ref{['eq:sigma']} are marked by dashed horizontal lines.
  • Figure 7.1: Localization errors of the SL-GFEM for multiple choices of $n$ and of the SLOD for a fixed coarse mesh.
  • Figure 7.2: Convergence plot of the SL-GFEM and SLOD for multiple choices of $n$ and $\ell$.
  • Figure 7.3: Coefficient $A_{\kappa}$ for $\kappa=10^4, 10^7$ (left and right).
  • Figure 7.4: Localization errors of the SL-GFEM and the SLOD for the high-contrast channeled coefficient $A_\kappa$ for $\kappa = 10^4, 10^7$ (left and right).
  • ...and 1 more figures

Theorems & Definitions (11)

  • Remark 5.1: Decay of $\sigma$
  • Theorem 5.3: A posteriori error bound
  • proof
  • Theorem 5.5: Bound on $d_n$
  • proof
  • Remark 5.6: Choice of parameters
  • Theorem 6.1: A priori error bound
  • proof
  • Theorem 6.2: Bound on $d_n$
  • proof
  • ...and 1 more