This paper aims at an accurate and efficient computation of effective quantities, e.g. the homogenized coefficients for approximating the solutions to partial differential equations with oscillatory coefficients. Typical multiscale methods are based on a micro–macro-coupling, where the macromodel describes the coarse scale behavior, and the micromodel is solved only locally to upscale the effective quantities, which are missing in the macromodel. The fact that the microproblems are solved over small domains within the entire macroscopic domain, implies imposing artificial boundary conditions on the boundary of the microscopic domains. A naive treatment of these artificial boundary conditions leads to a first-order error in 𝜀/δ, where 𝜀 < δ represents the characteristic length of the small scale oscillations and δd is the size of microdomain. This error dominates all other errors originating from the discretization of the macro and the microproblems, and its reduction is a main issue in today’s engineering multiscale computations. The objective of this work is to analyze a parabolic approach, first announced in A. Abdulle, D. Arjmand, E. Paganoni, C. R. Acad. Sci. Paris, Ser. I, 2019, for computing the homogenized coefficients with arbitrarily high convergence rates in 𝜀/δ. The analysis covers the setting of periodic microstructure, and numerical simulations are provided to verify the theoretical findings for more general settings, e.g. non-periodic microstructures.

中文翻译：

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数值均匀化共振误差指数衰减的抛物线局部问题

本文旨在准确有效地计算有效量，例如用于逼近具有振荡系数的偏微分方程解的均匀系数。典型的多尺度方法基于微观-宏观耦合，其中宏观模型描述了粗尺度行为，而微观模型仅在局部求解以放大宏观模型中缺失的有效量。在整个宏观域内的小域上解决微观问题的事实意味着在微观域的边界上施加人工边界条件。对这些人为边界条件的幼稚处理会导致一阶误差𝜀/δ， 在哪里𝜀 < δ表示小尺度振荡的特征长度和δd是微域的大小。该误差支配了源自宏观和微观问题离散化的所有其他误差，其减少是当今工程多尺度计算中的主要问题。这项工作的目的是分析抛物线​​方法，该方法首先在 A. Abdulle、D. Arjmand、E. Paganoni、CR学院。科学。巴黎，爵士。一世, 2019 年，用于计算具有任意高收敛速度的同质化系数𝜀/δ. 分析涵盖了周期性微观结构的设置，并提供了数值模拟来验证更一般设置的理论发现，例如非周期性微观结构。