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A Time Dependent Approach for Removing the Cell Boundary Error in Elliptic Homogenization Problems
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0002-6321-8619
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
(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: urn:nbn:se:kth:diva-129241OAI: oai:DiVA.org:kth-129241DiVA: diva2:651107
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
In thesis
1. Analysis and Applications of the Heterogeneous Multiscale Methods for Multiscale Elliptic and Hyperbolic Partial Differential Equations
Open this publication in new window or tab >>Analysis and Applications of the Heterogeneous Multiscale Methods for Multiscale Elliptic and Hyperbolic Partial Differential Equations
2013 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns the applications and analysis of the Heterogeneous Multiscale methods (HMM) for Multiscale Elliptic and Hyperbolic Partial Differential Equations. We have gathered the main contributions in two papers.

The first paper deals with the cell-boundary error which is present in multi-scale algorithms for elliptic homogenization problems. Typical multi-scale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. Solving the micro model requires, on the other hand, imposing boundary conditions on the boundary of the microscopic domain. Imposing a naive boundary condition leads to $O(\varepsilon/\eta)$ error in the computation, where $\varepsilon$ is the size of the microscopic variations in the media and $\eta$ is the size of the micro-domain. Until now, strategies were proposed to improve the convergence rate up to fourth-order in $\varepsilon/\eta$ at best. However, the removal of this error in multi-scale algorithms still remains an important open problem. In this paper, we present an approach with a time-dependent model which is general in terms of dimension. With this approach we are able to obtain $O((\varepsilon/\eta)^q)$ and $O((\varepsilon/\eta)^q  + \eta^p)$ convergence rates in periodic and locally-periodic media respectively, where $p,q$ can be chosen arbitrarily large.     

In the second paper, we analyze a multi-scale method developed under the Heterogeneous Multi-Scale Methods (HMM) framework for numerical approximation of wave propagation problems in periodic media. In particular, we are interested in the long time $O(\varepsilon^{-2})$ wave propagation. In the method, the microscopic model uses the macro solutions as initial data. In short-time wave propagation problems a linear interpolant of the macro variables can be used as the initial data for the micro-model. However, in long-time multi-scale wave problems the linear data does not suffice and one has to use a third-degree interpolant of the coarse data to capture the $O(1)$ dispersive effects apperaing in the long time. In this paper, we prove that through using an initial data consistent with the current macro state, HMM captures this dispersive effects up to any desired order of accuracy in terms of $\varepsilon/\eta$. We use two new ideas, namely quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic hyperbolic PDEs. As a byproduct, these ideas naturally reveal the role of consistency for high accuracy approximation of homogenized quantities.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. viii, 19 p.
Series
Trita-NA, ISSN 0348-2952 ; 13:02
Keyword
Multiscale Methods, HMM, Multiscale Wave Equation, Multiscale Elliptic Equation, Long Time Wave Propagation
National Category
Computational Mathematics
Research subject
SRA - E-Science (SeRC)
Identifiers
urn:nbn:se:kth:diva-129237 (URN)978-91-7501-884-3 (ISBN)
Presentation
2013-10-11, D42, Lindstedsvägen 5, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Swedish e‐Science Research Center, 649031
Note

QC 20130926

Available from: 2013-09-26 Created: 2013-09-24 Last updated: 2013-09-26Bibliographically approved
2. 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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Runborg, Olof

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