  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.
