On the nature of the boundary resonance error in numerical homogenization and its reduction

TL;DR

This work analyzes the boundary resonance error in numerical homogenization of multiscale elliptic problems and shows that naive microscale boundary conditions yield an $O(\epsilon/\eta)$ error that can overwhelm discretization. It introduces a novel averaging approach that does not modify the cell problem: solve the microscale problem on a sequence of domain sizes and weight the results with kernels having vanishing moments to accelerate convergence to the homogenized limit. The authors rigorously derive a 1D decomposition $E(\delta)=\frac{1}{\delta}P(\delta)+\frac{1}{\delta}\int \zeta$ and prove that, for appropriate kernels, the resonance error decays as $O((\epsilon/\eta)^{\min\{p+1,q+3\}})$ in 1D, with numerical evidence in 2D supporting similar behavior. The method preserves the original cell problem, is parallelizable, and can be integrated with reduced-basis techniques, offering practical improvements for high-dimensional multiscale simulations.

Abstract

Numerical homogenization of multiscale equations typically requires taking an average of the solution to a microscale problem. Both the boundary conditions and domain size of the microscale problem play an important role in the accuracy of the homogenization procedure. In particular, imposing naive boundary conditions leads to a $\mathcal{O}(ε/η)$ error in the computation, where $ε$ is the characteristic size of the microscopic fluctuations in the heterogeneous media, and $η$ is the size of the microscopic domain. This so-called boundary, or ``cell resonance" error can dominate discretization error and pollute the entire homogenization scheme. There exist several techniques in the literature to reduce the error. Most strategies involve modifying the form of the microscale cell problem. Below we present an alternative procedure based on the observation that the resonance error itself is an oscillatory function of domain size $η$. After rigorously characterizing the oscillatory behavior for one dimensional and quasi-one dimensional microscale domains, we present a novel strategy to reduce the resonance error. Rather than modifying the form of the cell problem, the original problem is solved for a sequence of domain sizes, and the results are averaged against kernels satisfying certain moment conditions and regularity properties. Numerical examples in one and two dimensions illustrate the utility of the approach.

Figures (7)

• Figure 1: The decay of $\eta^{r} f(\eta/\epsilon)$ and $\Upsilon_r(\eta)$ versus $\eta$ for various averaging kernels $K \in \mathbb{K}^{-p,q}$ of the form \ref{['eq:kernel_form']}. (Left) $r=2$ and $f(x)=1.1+\sin(2\pi x)$. (Right) $r=1$ and $f(x)$ is the even periodic extension of the piecewise constant function $\chi_{[0,1/4]} + \chi_{[1/4,1/2]}$.
• Figure 1: A two-dimensional "tubular" domain $I_{\delta} = [-\delta/2,\delta/2]\times[-1/2,1/2]$, $\delta \notin \mathbb{N}$.
• Figure 1: A comparison of the absolute value of the one-dimensional cell resonance error with the absolute value of the averaged error for various averaging kernels $K\in\mathbb{K}^{-p,q}$ for the three oscillatory functions $a^{\epsilon}$ given in \ref{['eq:1d_aeps_formulas']}.
• Figure 2: Visualization of the periodic oscillatory function \ref{['eq:aeps_case2']} (left) and corresponding components of the cell resonance error as a function of domain size (right).
• Figure 3: Visualization of the quasi-periodic oscillatory function \ref{['eq:aeps_case4']} (left) and corresponding components of the cell resonance error as a function of domain size (right).

Theorems & Definitions (11)

• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Lemma 2.4
• Lemma 3.1
• Definition 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Remark 3.6