The role of numerical integration in numerical homogenization

Abdulle, Assyr

doi:10.1051/proc/201550001

Cited by 3 publications

(2 citation statements)

References 34 publications

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“…As we have seen, the choice of a well-suited quadrature formula for FE-HMM is crucial, cf. [30]. Firstly, the quadrature must be accurate enough, such that (Q) holds.…”

Section: Practical Illustrationsmentioning

confidence: 99%

On the approximation of electromagnetic fields by edge finite elements. Part 2: A heterogeneous multiscale method for Maxwell’s equations

Ciarlet¹,

Fliss²,

Stohrer³

2017

Computers & Mathematics with Applications

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In the second part of this series of papers we consider highly oscillatory media. In this situation, the need for a triangulation that resolves all microscopic details of the medium makes standard edge finite elements impractical because of the resulting tremendous computational load. On the other hand, undersampling by using a coarse mesh might lead to inaccurate results. To overcome these difficulties and to improve the ratio between accuracy and computational costs, homogenization techniques can be used. In this paper we recall analytical homogenization results and propose a novel numerical homogenization scheme for Maxwell's equations in frequency domain. This scheme follows the design principles of heterogeneous multiscale methods. We prove convergence to the effective solution of the multiscale Maxwell's equations in a periodic setting and give numerical experiments in accordance to the stated results.

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“…As we have seen, the choice of a well-suited quadrature formula for FE-HMM is crucial, cf. [30]. Firstly, the quadrature must be accurate enough, such that (Q) holds.…”

Section: Practical Illustrationsmentioning

confidence: 99%

On the approximation of electromagnetic fields by edge finite elements. Part 2: A heterogeneous multiscale method for Maxwell’s equations

Ciarlet¹,

Fliss²,

Stohrer³

2017

Computers & Mathematics with Applications

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“…We assume that these formulas satisfy the requirements that ensure the optimal convergence rates of the FEM with numerical quadrature (see [21,20] or [1]). For every macro element K ∈ T H and every j ∈ {1, .…”

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confidence: 99%

Effective Models for Long Time Wave Propagation in Locally Periodic Media

Abdulle¹,

Pouchon²

2018

SIAM J. Numer. Anal.

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Abstract. A family of effective equations for the wave equation in locally periodic media over long time is derived. In particular, explicit formulas for the effective tensors are provided. To validate the derivation, an a priori error estimate between the effective solutions and the original wave is proved. As the dependence of the estimate on the domain is explicit, the result holds in arbitrarily large periodic hypercube. This constitutes the first analysis for the description of long time effects for the wave equation in locally periodic media. Thanks to this result, the long time a priori error analysis of the numerical homogenization method presented in [A. Abdulle and T. Pouchon, SIAM J. Numer. Anal., 54, 2016Anal., 54, , pp. 1507Anal., 54, -1534 is generalized to the case of a locally periodic tensor.

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Error estimates for finite element approximations of nonlinear monotone elliptic problems with application to numerical homogenization

Abdulle

Huber

2016

Numer. Methods Partial Differential Eq.

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We propose a multiscale method based on a finite element heterogeneous multiscale method (in space) and the implicit Euler integrator (in time) to solve nonlinear monotone parabolic problems with multiple scales due to spatial heterogeneities varying rapidly at a microscopic scale. The multiscale method approximates the homogenized solution at computational cost independent of the small scale by performing numerical upscaling (coupling of macro and micro finite element methods). Taking into account the error due to time discretization as well as macro and micro spatial discretizations, the convergence of the method is proved in the general L p (W 1,p ) setting. For p = 2, optimal convergence rates in the L 2 (H 1 ) and C 0 (L 2 ) norm are derived. Numerical experiments illustrate the theoretical error estimates and the applicability of the multiscale method to practical problems.

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The role of numerical integration in numerical homogenization

Cited by 3 publications

References 34 publications

On the approximation of electromagnetic fields by edge finite elements. Part 2: A heterogeneous multiscale method for Maxwell’s equations

On the approximation of electromagnetic fields by edge finite elements. Part 2: A heterogeneous multiscale method for Maxwell’s equations

Effective Models for Long Time Wave Propagation in Locally Periodic Media

Error estimates for finite element approximations of nonlinear monotone elliptic problems with application to numerical homogenization

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