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.

翻译：多尺度方程的数值均匀化通常需要计算微观问题解的平均值。微观问题的边界条件和区域大小对均匀化过程的精度起着重要作用。特别地，施加朴素边界条件会导致计算中出现$\mathcal{O}(\epsilon/\eta)$量级的误差，其中$\epsilon$是异质介质中微观波动的特征尺度，$\eta$是微观区域的尺度。这种所谓的边界或“单元共振”误差可能支配离散误差，并污染整个均匀化方案。文献中存在几种减小该误差的技术，大多涉及修改微观单元问题的形式。下面我们提出一种替代方法，其基于共振误差本身是区域大小$\eta$的振荡函数这一观察。在严格刻画一维和准一维微观区域振荡行为的基础上，我们提出了一种新颖的减小共振误差的策略。该方法不修改单元问题的形式，而是针对一系列区域大小求解原始问题，并将结果与满足特定矩条件和正则性性质的核函数进行平均处理。一维和二维数值算例验证了该方法的有效性。