We propose a multiscale model reduction method for partial differential equations. Themain purpose of this method is to derive an effective equation for multiscale problemswithout scale separation. An essential ingredient of our method is to decompose theharmonic coordinates into a smooth part and a highly oscillatory part so that the smoothpart is invertible and the highly oscillatory part is small. Such a decomposition plays akey role in our construction of the effective equation. We show that the solution to theeffective equation is in H2, and can be approximated by a regularcoarse mesh. When the multiscale problem has scale separation and a periodic structure,our method recovers the traditional homogenized equation. Furthermore, we provide erroranalysis for our method and show that the solution to the effective equation is close tothe original multiscale solution in the H1 norm. Numerical results are presentedto demonstrate the accuracy and robustness of the proposed method for several multiscaleproblems without scale separation, including a problem with a high contrastcoefficient.
