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A parabolic local problem with exponential decay of the resonance error for numerical homogenization
arXiv - CS - Numerical Analysis Pub Date : 2020-01-15 , DOI: arxiv-2001.05543
Assyr Abdulle and Doghonay Arjmand and Edoardo Paganoni

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 macro model describes the coarse scale behaviour, and the micro model is solved only locally to upscale the effective 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 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 $\varepsilon/\delta$, where $\varepsilon < \delta$ represents the characteristic length of the small scale oscillations and $\delta^d$ is the size of micro domain. This error dominates all other errors originating from the discretization of the macro and the micro problems, and its reduction is a main issue in today's engineering multiscale computations. The objective of the present 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 $\varepsilon/\delta$. The analysis covers the setting of periodic micro structure, and numerical simulations are provided to verify the theoretical findings for more general settings, e.g. random stationary micro structures.

中文翻译:

数值均匀化的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齐次系数。典型的多尺度方法基于微观-宏观耦合,其中宏观模型描述粗尺度行为,微观模型仅在局部求解以放大宏观模型中缺失的有效量。微观问题在整个宏观域内的小域上解决这一事实意味着在微观域的边界上强加人为边界条件。对这些人工边界条件的幼稚处理会导致 $\varepsilon/\delta$ 中的一阶错误,其中 $\varepsilon < \delta$ 代表小尺度振荡的特征长度,$\delta^d$ 是微域的大小。这个误差在所有其他源于宏观和微观问题离散化的误差中占主导地位,它的减少是当今工程多尺度计算中的一个主要问题。当前工作的目标是分析抛物线​​方法,该方法首先在 [A. Abdulle, D. Arjmand, E. Paganoni, CR Acad。科学。巴黎,爵士。I, 2019],用于计算 $\varepsilon/\delta$ 中具有任意高收敛率的均质系数。分析涵盖周期性微观结构的设置,并提供数值模拟以验证更一般设置的理论发现,例如随机静止微观结构。这个误差在所有其他源于宏观和微观问题离散化的误差中占主导地位,它的减少是当今工程多尺度计算中的一个主要问题。当前工作的目标是分析抛物线​​方法,该方法首先在 [A. Abdulle, D. Arjmand, E. Paganoni, CR Acad。科学。巴黎,爵士。I, 2019],用于计算 $\varepsilon/\delta$ 中具有任意高收敛率的均质系数。分析涵盖周期性微观结构的设置,并提供数值模拟以验证更一般设置的理论发现,例如随机静止微观结构。该误差主导了源自宏观和微观问题的离散化的所有其他误差,并且其减少是当今工程多尺度计算中的主要问题。当前工作的目标是分析抛物线​​方法,该方法首先在 [A. Abdulle, D. Arjmand, E. Paganoni, CR Acad。科学。巴黎,爵士。I, 2019],用于在 $\varepsilon/\delta$ 中计算具有任意高收敛率的均质系数。分析涵盖周期性微观结构的设置,并提供数值模拟以验证更一般设置的理论发现,例如随机静止微观结构。当前工作的目标是分析抛物线​​方法,该方法首先在 [A. Abdulle, D. Arjmand, E. Paganoni, CR Acad。科学。巴黎,爵士。I, 2019],用于计算 $\varepsilon/\delta$ 中具有任意高收敛率的均质系数。分析涵盖周期性微观结构的设置,并提供数值模拟以验证更一般设置的理论发现,例如随机静止微观结构。当前工作的目标是分析抛物线​​方法,该方法首先在 [A. Abdulle, D. Arjmand, E. Paganoni, CR Acad。科学。巴黎,爵士。I, 2019],用于在 $\varepsilon/\delta$ 中计算具有任意高收敛率的均质系数。分析涵盖周期性微观结构的设置,并提供数值模拟以验证更一般设置的理论发现,例如随机静止微观结构。
更新日期:2020-01-17
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