  Numerical multiscale methods usually rely on some coupling between a
macroscopic and a microscopic model. The macroscopic model is incomplete as
effective quantities, such as the homogenized material coefficients or fluxes,
are missing in the model. These effective data need to be computed by running
local microscale simulations followed by a local averaging of the microscopic
information. Motivated by the classical homogenization theory, it is a common
practice to use local elliptic cell problems for computing the missing
homogenized coefficients in the macro model. Such a consideration results in a
first order error $O(\varepsilon/\delta)$, where $\varepsilon$ represents the
wavelength of the microscale variations and $\delta$ is the size of the
microscopic simulation boxes. This error, called "resonance error", originates
from the boundary conditions used in the micro-problem and typically dominates
all other errors in a multiscale numerical method. Optimal decay of the
resonance error remains an open problem, although several interesting
approaches reducing the effect of the boundary have been proposed over the last
two decades. In this paper, as an attempt to resolve this problem, we propose a
computationally efficient, fully elliptic approach with exponential decay of
the resonance error.
