An elliptic local problem with exponential decay of the resonance error for numerical homogenization

01/17/2020 ∙ by Assyr Abdulle, et al. ∙ 0

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(ε/δ), where ε represents the wavelength of the microscale variations and δ is the size of the microscopic simulation boxes. This error, called "resonance error", originates from the boundary conditions used in the micro-problem 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.

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1 Introduction

1.1 Motivation - source of the boundary error

1.2 Existing approaches to reduce the resonance error

1.3 Notations and definitions

2 A modified elliptic approach

2.1 Relation with parabolic cell problems

3 Exponential decay of the resonance error for the modified elliptic approach

4 Approximation of the exponential operator

4.1 Spectral truncation

4.2 Approximation by the Arnoldi method

4.3 Approximation of the cell problem and computational cost

5 Numerical tests

Acknowledgments

The authors are grateful to Stefano Massei and Kathryn Lund for helpful discussion. This research is partially supported by Swiss National Science Foundation, grant no. .

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