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Hybrid Localized Spectral Decomposition for multiscale problems

Alexandre L. Madureira, Marcus Sarkis

TL;DR

This work introduces a dimensionally robust, hybridized multiscale method for elliptic problems with heterogeneous and potentially high-contrast coefficients. It recasts the problem through a primal hybrid formulation and leverages a space-decomposition to reduce the global coupling to localized elliptic solves, with exponential decay of the nonlocal components. To handle high contrast, it enriches the local spaces via edge-based generalized eigenproblems, forming a Localized Spectral Decomposition (LSD) that yields contrast-independent decay and optimal a priori error bounds; in low-contrast regimes the method simplifies while preserving exponential localization. A finite-dimensional right-hand side and a practical, scalable algorithm (with pre-processing) enable efficient computation, and numerical experiments demonstrate robust accuracy and significant localization benefits across smooth, oscillatory, and highly contrasted coefficients, with applicability to elasticity as well.

Abstract

We consider a finite element method for elliptic equation with heterogeneous and possibly high-contrast coefficients based on primal hybrid formulation. A space decomposition as in FETI and BDCC allows a sequential computations of the unknowns through elliptic problems and satisfies equilibrium constraints. One of the resulting problems is non-local but with exponentially decaying solutions, enabling a practical scheme where the basis functions have an extended, but still local, support. We obtain quasi-optimal a priori error estimates for low-contrast problems assuming minimal regularity of the solutions. To also consider the high-contrast case, we propose a variant of our method, enriching the space solution via local eigenvalue problems and obtaining optimal a priori error estimate that mitigates the effect of having coefficients with different magnitudes and again assuming no regularity of the solution. The technique developed is dimensional independent and easy to extend to other problems such as elasticity.

Hybrid Localized Spectral Decomposition for multiscale problems

TL;DR

This work introduces a dimensionally robust, hybridized multiscale method for elliptic problems with heterogeneous and potentially high-contrast coefficients. It recasts the problem through a primal hybrid formulation and leverages a space-decomposition to reduce the global coupling to localized elliptic solves, with exponential decay of the nonlocal components. To handle high contrast, it enriches the local spaces via edge-based generalized eigenproblems, forming a Localized Spectral Decomposition (LSD) that yields contrast-independent decay and optimal a priori error bounds; in low-contrast regimes the method simplifies while preserving exponential localization. A finite-dimensional right-hand side and a practical, scalable algorithm (with pre-processing) enable efficient computation, and numerical experiments demonstrate robust accuracy and significant localization benefits across smooth, oscillatory, and highly contrasted coefficients, with applicability to elasticity as well.

Abstract

We consider a finite element method for elliptic equation with heterogeneous and possibly high-contrast coefficients based on primal hybrid formulation. A space decomposition as in FETI and BDCC allows a sequential computations of the unknowns through elliptic problems and satisfies equilibrium constraints. One of the resulting problems is non-local but with exponentially decaying solutions, enabling a practical scheme where the basis functions have an extended, but still local, support. We obtain quasi-optimal a priori error estimates for low-contrast problems assuming minimal regularity of the solutions. To also consider the high-contrast case, we propose a variant of our method, enriching the space solution via local eigenvalue problems and obtaining optimal a priori error estimate that mitigates the effect of having coefficients with different magnitudes and again assuming no regularity of the solution. The technique developed is dimensional independent and easy to extend to other problems such as elasticity.

Paper Structure

This paper contains 14 sections, 15 theorems, 159 equations, 6 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Continuous Problem using Hybrid Formulation
  3. Hybrid Localized Finite Elements
  4. Decaying High-Contrast
  5. Decaying Low-Contrast
  6. Finite Dimensional Right-Hand Side
  7. Algorithms
  8. Numerical Examples
  9. Exact Solution available
  10. Oscillatory kernel
  11. Beanbag test
  12. High-contrast channels
  13. Auxiliary results
  14. Notations

Key Result

Lemma 1

Let $F$ be a face of ${\partial\tau}$ shared by the elements $\tau$, $\tau^\prime\in{\mathcal{T}_H}$. If $F$ is on the boundary $\partial\Omega$, assume $\tau^\prime=\emptyset$. Then for all $\tilde{\mu}_h^\tau=\{\tilde{\mu}_h^{F,\triangle},\tilde{\mu}_h^{F_\tau^c}\}\in\tilde{\Lambda}_h^{F,\triangle}\times\tilde{\Lambda}_h^{F_\tau^c}$ and all $\tilde{\mu}_h^{\tau^\prime}=\{\tilde{\mu}_h^{F,\trian

Figures (6)

  • Figure 1: Mesh and submesh employed in all numerical tests. We consider $H=1/8$ and $h\approx H/9$.
  • Figure 2: Test \ref{['ss:exact']}: Log-plot of energy decay in the case of empty $\tilde{\Lambda}_h^\Pi$.
  • Figure 3: Test \ref{['ss:osc']}: Eigenvalues for problems I, II.
  • Figure 4: Test \ref{['ss:beabag']}. Top: Exact solution and LOD with $J=1$. Bottom: LSD solutions with $J=1$ and $\alpha_{\rm{stab}}=3.0$ and $\alpha_{\rm{stab}}=1.7$.
  • Figure 5: Test \ref{['ss:beabag']}: Relative energy error with respect to contrast (=$1/a_{\text{min}}$). The blue curve corresponds to LOD with $J=1$ and the red curve is related to the LSD method with $J=1$ and $\alpha_{\rm{stab}}=1.7$
  • ...and 1 more figures

Theorems & Definitions (42)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 3
  • Theorem 3
  • proof
  • Lemma 4
  • ...and 32 more