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

TL;DR

This paper investigates the boundary resonance error in numerical homogenization, characterizes its oscillatory nature, and introduces a novel averaging method over domain sizes to reduce this error without modifying the cell problem.

Contribution

The authors provide a new understanding of the boundary resonance error as an oscillatory function and propose an averaging technique to mitigate it, avoiding modifications to the cell problem.

Findings

Resonance error oscillates with domain size in homogenization.

Averaging over domain sizes effectively reduces the boundary resonance error.

Numerical examples demonstrate the method's effectiveness in 1D and 2D cases.

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 $\mathcal{O}(\epsilon/\eta)$ error in the computation, where $\epsilon$ is the characteristic size of the microscopic fluctuations in the heterogeneous media, and $\eta$ 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 $\eta$. 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.