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Publication# MATHICSE technical Report : A parabolic local problem with exponential decay of the resonance error for numerical homogenization

Abstract

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.

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Multiscale problems, such as modelling flows through porous media or predicting the mechanical properties of composite materials, are of great interest in many scientific areas. Analytical models describing these phenomena are rarely available, and one must recur to numerical simulations. This represents a great computational challenge, because of the prohibitive computational cost of resolving the small scales. Multiscale numerical methods are therefore necessary to solve multiscale problems within reasonable computational time and resources. In particular, numerical homogenization techniques aim to capture the macroscopic behaviour with equations whose coefficients are computed numerically from the solutions of corrector problems at the microscale. A lack of knowledge of the coupling conditions between the micro- and the macro-scales brings in the so-called resonance error, which affects the accuracy of all multiscale methods. This source of error often dominates the numerical discretization errors and increasing its rate of decay is crucial for improving the accuracy of multiscale methods.
In this work, we propose two novel upscaling schemes with arbitrarily high convergence rates of the resonance error to approximate the homogenized coefficients of scalar, linear second order elliptic differential equations. The first one is based on a parabolic equation, inspired by a model employed to compute the effective diffusive coefficients in stochastic diffusion processes. By using the approximation properties of smooth filtering functions, the homogenized coefficients can be approximated with arbitrary rates of accuracy. This claim is proved through an a priori convergence analysis, under the assumption of smooth periodic multiscale coefficients. Numerical experiments verify the expected convergence rates also under more general assumptions, such as discontinuous and random coefficients. The second method originates from the first by integrating the parabolic equation over a finite time interval. This method is referred to as the modified elliptic approach, because of the presence of a right-hand side which can be interpreted in terms of continuous semigroups and can be approximated numerically by Krylov subspace methods. The same convergence results as in the parabolic approach hold true. As a last step, a convergence analysis of the resonance error for the modified elliptic approach in the context of equations with random coefficients is performed. In this case, the resonance error is composed of a variance and a bias term, which can be bounded from above by a function decaying to zero. Numerical experiments reveal that the convergence rate of the resonance error for random coefficients is hampered, in comparison to the case of periodic coefficients, but the modified elliptic approach nevertheless outperforms standard methods.

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.