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This paper aims at an accurate and ecient computation of eective quantities, e.g., the homogenized coecients for approximating the solu- tions to partial dierential equations with oscillatory coecients. Typical multiscale methods are based on a micro-macro coupling, where the macro model describes the coarse scale behaviour, and the micro model is solved only locally to upscale the eective quantities, which are missing in the macro model. The fact that the micro problems are solved over small domains within the entire macroscopic domain, implies imposing arti- cial boundary conditions on the boundary of the microscopic domains. A naive treatment of these articial boundary conditions leads to a rst order error in "=, where " < represents the characteristic length of the small scale oscillations and d is the size of micro domain. This er- ror dominates all other errors originating from the discretization of the macro and the micro problems, and its reduction is a main issue in to- day's engineering multiscale computations. The objective of the present work is to analyse a parabolic approach, rst announced in [A. Abdulle, D. Arjmand, E. Paganoni, C. R. Acad. Sci. Paris, Ser. I, 2019], for computing the homogenized coecients with arbitrarily high convergence rates in "=. The analysis covers the setting of periodic microstructure, and numerical simulations are provided to verify the theoretical ndings for more general settings, e.g. random stationary micro structures.