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Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems

Assyr Abdulle, Doghonay Arjmand, Edoardo Paganoni

TL;DR

The paper tackles resonance error in numerical homogenization of multiscale elliptic PDEs by introducing two strategies: a parabolic cell-problem formulation and a modified elliptic upscaling with spectral correction. It proves an equivalence between the classical elliptic and the new parabolic formulation, and provides explicit exponential-convergence results for both approaches under suitable parameter choices, including filter smoothness and time horizons. Numerical validation in a 2D periodic setting demonstrates that the boundary-error decays exponentially with moderate coarse-domain sizes, while maintaining comparable computational cost to standard methods. These approaches offer high-accuracy upscaling without the usual $O(1/R)$ resonance limitations, enabling efficient upscaling for periodic, quasi-periodic, or stochastic microstructures with improved practical impact.

Abstract

This paper presents two new approaches for finding the homogenized coefficients of multiscale elliptic PDEs. Standard approaches for computing the homogenized coefficients suffer from the so-called resonance error, originating from a mismatch between the true and the computational boundary conditions. Our new methods, based on solutions of parabolic and elliptic cell-problems, result in an exponential decay of the resonance error.

Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems

TL;DR

The paper tackles resonance error in numerical homogenization of multiscale elliptic PDEs by introducing two strategies: a parabolic cell-problem formulation and a modified elliptic upscaling with spectral correction. It proves an equivalence between the classical elliptic and the new parabolic formulation, and provides explicit exponential-convergence results for both approaches under suitable parameter choices, including filter smoothness and time horizons. Numerical validation in a 2D periodic setting demonstrates that the boundary-error decays exponentially with moderate coarse-domain sizes, while maintaining comparable computational cost to standard methods. These approaches offer high-accuracy upscaling without the usual resonance limitations, enabling efficient upscaling for periodic, quasi-periodic, or stochastic microstructures with improved practical impact.

Abstract

This paper presents two new approaches for finding the homogenized coefficients of multiscale elliptic PDEs. Standard approaches for computing the homogenized coefficients suffer from the so-called resonance error, originating from a mismatch between the true and the computational boundary conditions. Our new methods, based on solutions of parabolic and elliptic cell-problems, result in an exponential decay of the resonance error.

Paper Structure

This paper contains 9 sections, 4 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. New algorithms for computing the homogenized tensor
  3. A parabolic approach
  4. A modified elliptic approach
  5. Main results
  6. Equivalence between the standard elliptic and the parabolic formulations
  7. Exponential convergence of the parabolic approach \ref{['eq:new a0 filtered']}
  8. Exponential convergence of the modified elliptic approach \ref{['eq:new_elliptic_upscaling_formula']}
  9. Numerical validation

Key Result

Theorem 1

Let $a \in \mathcal{M}(\alpha,\beta,K_R)$ and let $\nabla\cdot \left(a\mathbf{e}_k\right) \in L^2(K_R)$, for $k =1,\dots,d$. Let $u_R^k\in X_0(\mathop{\mathrm{\mathbb{R}}}\nolimits_+,K_R)$ be the unique weak solution of eq:parabolic dirichlet problem and $\psi^k_{R} \in H^1_0( K_{R} )$ be the unique

Figures (1)

  • Figure 1: Modelling error for the homogenization of the multiscale coefficients \ref{['eq:tensor exp']}. Modelling parameters are $k_o = 1/2$ and $k_T = \frac{k_{o}}{\pi \sqrt{4 \beta \alpha}}$.

Theorems & Definitions (6)

  • Definition 1
  • Theorem 1
  • proof
  • Corollary 1
  • Theorem 2
  • Theorem 3