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Analysis and Applications of the Heterogeneous Multiscale Methods for Multiscale Elliptic and Hyperbolic Partial Differential Equations
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
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 [en]
Multiscale Methods, HMM, Multiscale Wave Equation, Multiscale Elliptic Equation, Long Time Wave Propagation
##### National Category
Computational Mathematics
##### Research subject
SRA - E-Science (SeRC)
##### Identifiers
ISBN: 978-91-7501-884-3OAI: oai:DiVA.org:kth-129237DiVA: diva2:651118
##### Presentation
2013-10-11, D42, Lindstedsvägen 5, KTH, Stockholm, 10:00 (English)
##### 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
##### List of papers
1. 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
##### 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
2. 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 (ScopusID)
##### 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: 2015-02-17Bibliographically approved

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