We propose a multiscale model reduction method for partial differential equations. The
main purpose of this method is to derive an effective equation for multiscale problems
without scale separation. An essential ingredient of our method is to decompose the
harmonic coordinates into a smooth part and a highly oscillatory part so that the smooth
part is invertible and the highly oscillatory part is small. Such a decomposition plays a
key role in our construction of the effective equation. We show that the solution to the
effective equation is in H2, and can be approximated by a regular
coarse mesh. When the multiscale problem has scale separation and a periodic structure,
our method recovers the traditional homogenized equation. Furthermore, we provide error
analysis for our method and show that the solution to the effective equation is close to
the original multiscale solution in the H1 norm. Numerical results are presented
to demonstrate the accuracy and robustness of the proposed method for several multiscale
problems without scale separation, including a problem with a high contrast
coefficient.