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Person# Edoardo Paganoni

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Exponential decay

A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where

Coefficient

In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or an expression. It may be a number (dimensionless), in which case it is known as a numeric

Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.
The function is often thought of as

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Doghonay Arjmand, Edoardo Paganoni

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.

Assyr Abdulle, Doghonay Arjmand, Edoardo Paganoni

Numerical multiscale methods usually rely on some coupling between a macroscopic and a microscopic model. The macroscopic model is incomplete as effective quantities, such as the homogenized material coefficients or fluxes, are missing in the model. These effective data need to be computed by running local microscale simulations followed by a local averaging of the microscopic information. Motivated by the classical homogenization theory, it is a common practice to use local elliptic cell problems for computing the missing homogenized coefficients in the macro model. Such a consideration results in a first order error O(E/8), where E represents the wavelength of the microscale variations and 8 is the size of the microscopic simulation boxes. This error, called ``resonance error,"" originates from the boundary conditions used in the microproblem and typically dominates all other errors in a multiscale numerical method. Optimal decay of the resonance error remains an open problem, although several interesting approaches reducing the effect of the boundary have been proposed over the last two decades. In this paper, as an attempt to resolve this problem, we propose a computationally efficient, fully elliptic approach with exponential decay of the resonance error.

Assyr Abdulle, Doghonay Arjmand, 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 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 epsilon/delta, where epsilon < delta represents the characteristic length of the small scale oscillations and delta(d) 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 epsilon/delta. 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.