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A Finite Element Heterogenous Multiscale Method with Improved Control Over the Modeling Error
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
ENSTA ParisTech.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Multiscale partial dierential equations (PDEs) are difficult to solve by traditional numerical methods due to the need to resolve the small wavelengths in the media over the entire computational domain. We develop and analyze a Finite Element Heterogeneous Multiscale Method (FE-HMM) for approximating the homogenized solutions of multiscale PDEs of elliptic, parabolic,and hyperbolic type. Typical multiscale methods require a coupling between a micro and a macromodel. Inspired from the homogenization theory, traditional FE-HMM schemes use elliptic PDEs as the micro model. We use, however, the second order wave equation as our micro model independent of the type of the problem on the macro level. This allows us to control the modeling error originating by the coupling between the dierent scales. In a spatially fully discrete a priori error analysis we prove that the modeling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media. We provide numerical examples in one and two dimensions confirming the theoretical results. Further examples show that the method captures the effective solutions in general non-periodic settings as well

National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-160120OAI: oai:DiVA.org:kth-160120DiVA: diva2:788650
Funder
Swedish e‐Science Research Center
Note

QS 2015

Available from: 2015-02-16 Created: 2015-02-16 Last updated: 2015-02-17Bibliographically approved
In thesis
1. Analysis and Applications of Heterogeneous Multiscale Methods for Multiscale Partial Differential Equations
Open this publication in new window or tab >>Analysis and Applications of Heterogeneous Multiscale Methods for Multiscale Partial Differential Equations
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis centers on the development and analysis of numerical multiscale methods for multiscale problems arising in steady heat conduction, heat transfer and wave propagation in heterogeneous media. In a multiscale problem several scales interact with each other to form a system which has variations over a wide range of scales. A direct numerical simulation of such problems requires resolving the small scales over a computational domain, typically much larger than the microscopic scales. This demands a tremendous computational cost. We develop and analyse multiscale methods based on the heterogeneous multiscale methods (HMM) framework, which captures the macroscopic variations in the solution at a cost much lower than traditional numerical recipes. HMM assumes that there is a macro and a micro model which describes the problem. The micro model is accurate but computationally expensive to solve. The macro model is inexpensive but incomplete as it lacks certain parameter values. These are upscaled by solving the micro model locally in small parts of the domain. The accuracy of the method is then linked to how accurately this upscaling procedure captures the right macroscopic effects. In this thesis we analyse the upscaling error of existing multiscale methods and also propose a micro model which significantly reduces the upscaling error invarious settings. In papers I and IV we give an analysis of a finite difference HMM (FD-HMM) for approximating the effective solutions of multiscale wave equations over long time scales. In particular, we consider time scales T^ε = O(ε−k ), k =1, 2, where ε represents the size of the microstructures in the medium. In this setting, waves exhibit non-trivial behaviour which do not appear over short time scales. We use new analytical tools to prove that the FD-HMM accurately captures the long time effects. We first, in Paper I, consider T^ε =O(ε−2 ) and analyze the accuracy of FD-HMM in a one-dimensional periodicsetting. The core analytical ideas are quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic wave equations.The analysis naturally reveals the role of consistency in HMM for high order approximation of effective quantities over long time scales. Next, in paperIV, we consider T^ε = O(ε−1 ) and use the tools in a multi-dimensional settingto analyze the accuracy of the FD-HMM in locally-periodic media where fast and slow variations are allowed at the same time. Moreover, in papers II and III we propose new multiscale methods which substantially improve the upscaling error in multiscale elliptic, parabolic and hyperbolic partial differential equations. In paper II we first propose a FD-HMM for solving elliptic homogenization problems. The strategy is to use the wave equation as the micro model even if the macro problem is of elliptic type. Next in paper III, we use this idea in a finite element HMM setting and generalize the approach to parabolic and hyperbolic problems. In a spatially fully discrete a priori error analysis we prove that the upscaling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2015. ix, 45 p.
Series
TRITA-MAT-A, 2015:03
Keyword
Numerical homogenization, long time wave propagation, multiscale PDEs
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-160122 (URN)978-91-7595-446-2 (ISBN)
Public defence
2015-03-06, D3, Lindstedsvägen 5, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Projects
Multiscale methods for wave propagation
Funder
Swedish e‐Science Research Center
Note

QC 20150216

Available from: 2015-02-16 Created: 2015-02-16 Last updated: 2015-02-17Bibliographically approved

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