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Analysis and Applications of Heterogeneous Multiscale Methods for Multiscale Partial Differential Equations
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
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 [en]
Numerical homogenization, long time wave propagation, multiscale PDEs
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-160122ISBN: 978-91-7595-446-2 (print)OAI: oai:DiVA.org:kth-160122DiVA: diva2:788665
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
List of papers
1. Analysis of heterogeneous multiscale methods for long time wave propagation problems
Open this publication in new window or tab >>Analysis of heterogeneous multiscale methods for long time wave propagation problems
2014 (English)In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 12, no 3, 1135-1166 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we analyze a multiscale method developed under the heterogeneous multiscale method (HMM) framework for numerical approximation of multiscale wave propagation problems in periodic media. In particular, we are interested in the long time O(epsilon(-2)) wave propagation, where e represents the size of the microscopic variations in the media. In large time scales, the solutions of multiscale wave equations exhibit O(1) dispersive effects which are not observed in short time scales. A typical HMM has two main components: a macromodel and a micromodel. The macromodel is incomplete and lacks a set of local data. In the setting of multiscale PDEs, one has to solve for the full oscillatory problem over local microscopic domains of size eta = O(epsilon) to upscale the parameter values which are missing in the macroscopic model. In this paper, we prove that if the microproblems are consistent with the macroscopic solutions, the HMM approximates the unknown parameter values in the macromodel up to any desired order of accuracy in terms of epsilon/eta..

Keyword
multiscale wave equation, long time wave equation, homogenization
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-129245 (URN)10.1137/140957573 (DOI)000343130500008 ()2-s2.0-84907940927 (Scopus ID)
Funder
Swedish e‐Science Research Center, 649031
Note

QC 20130924. Updated from manuscript to article in journal.

Available from: 2013-09-24 Created: 2013-09-24 Last updated: 2017-12-06Bibliographically approved
2. A Time Dependent Approach for Removing the Cell Boundary Error in Elliptic Homogenization Problems
Open this publication in new window or tab >>A Time Dependent Approach for Removing the Cell Boundary Error in Elliptic Homogenization Problems
(English)Manuscript (preprint) (Other academic)
Abstract [en]

This paper concerns the cell-boundary error present in multiscale algorithms for elliptichomogenization problems. Typical multiscale methods have two essential components: amacro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. To solve the micro model, boundary conditions are required on the boundary of the microscopic domain. Imposing a naive boundary condition leads to O(e/eta) error in the computation, where e is the size of the microscopic variations in the media and eta is the size of the micro-domain. The removal of this error in modern multiscale algorithms still remains an important open problem. In this paper, we present a time-dependent approach which is general in terms of dimension. We provide a theorem which shows that we have arbitrarily high order convergence rates in terms of e/eta in theperiodic setting. Additionally, we present numerical evidence showing that the method improves the O(e/eta) error to O(e) in general non-periodic media.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-129241 (URN)
Funder
Swedish e‐Science Research Center, 649031
Note

QC 20130924

Available from: 2013-09-24 Created: 2013-09-24 Last updated: 2015-02-17Bibliographically approved
3. A Finite Element Heterogenous Multiscale Method with Improved Control Over the Modeling Error
Open this publication in new window or tab >>A Finite Element Heterogenous Multiscale Method with Improved Control Over the Modeling Error
(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:nbn:se:kth:diva-160120 (URN)
Funder
Swedish e‐Science Research Center
Note

QS 2015

Available from: 2015-02-16 Created: 2015-02-16 Last updated: 2015-02-17Bibliographically approved
4. Analysis of HMM for Long Time Multiscale Wave Propagation Problems in Locally-Periodic Media
Open this publication in new window or tab >>Analysis of HMM for Long Time Multiscale Wave Propagation Problems in Locally-Periodic Media
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Multiscale wave propagation problems are difficult to solve numerically due to the interaction of different scales inherent in the problem. Extracting information about the average behaviour of the system requires resolving small scales in the problem. This leads to a tremendous computational burden if the size of microscopic variations are much smaller than the size of scales of interest. Heterogeneous multiscale methods (HMM) is a tool to avoid resolving the small scales everywhere. Nevertheless, it approximates the average part of the solution by upscaling the microscopic information on a small part of the domain. This leads to a substantial improvement in the computational cost. In this article, we analyze an HMM-based numerical method which approximates the long time behaviour of multiscale wave equations. In particular, we consider theoretically challenging case of locally-periodic media where fast and slow variations are allowed at the same time. We are interested in the long time regime (T=O(e^{-1})), where e represents the wavelength of the fast variations in themedia. We first use asymptotic expansions to derive effective equations describing the long time effects of the multiscale waves in multi-dimensional locally-periodic media. We then show that HMM captures these non-trivial long time eects. All the theoretical statements are general in terms of dimension. Two dimensional numericale xamples are considered to support our theoretical arguments

National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-160121 (URN)
Projects
Multiscale methods for wave propagation
Funder
Swedish e‐Science Research Center
Note

QS 2015

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

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