Multiscale Analytical Solutions and Homogenization of n-Dimensional Generalized Elliptic Equations

Persistent Link:
http://hdl.handle.net/10150/194912
Title:
Multiscale Analytical Solutions and Homogenization of n-Dimensional Generalized Elliptic Equations
Author:
Sviercoski, Rosangela
Issue Date:
2005
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
In this dissertation, we present multiscale analytical solutions, in the weak sense, to the generalized Laplace's equation in Ω ⊂ Rⁿ, subject to periodic and nonperiodic boundary conditions. They are called multiscale solutions since they depend on a coefficient which takes a wide possible range of scales. We define forms of nonseparable coefficient functions in Lᵖ(Ω) such that the solutions are valid for the periodic and nonperiodic cases. In the periodic case, one such solution corresponds to the auxiliary cell problem in homogenization theory. Based on the proposed analytical solution, we were able to write explicitly the analytical form for the upscaled equation with an effective coefficient, for linear and nonlinear cases including the one with body forces. This was done by performing the two-scale asymptotic expansion for linear and nonlinear equations in divergence form with periodic coefficient. We proved that the proposed homogenized coefficient satisfies the Voigt-Reiss inequality. By performing numerical experiments and error analyses, we were able to compare the heterogeneous equation and its homogenized approximation in order to define criteria in terms of allowable heterogeneity in the domain to obtain a good approximation. The results presented, in this dissertation, have laid mathematical groundwork to better understand and apply multiscale processes under a deterministic point of view.
Type:
text; Electronic Dissertation
Keywords:
Multiscale Analytical Solutions; Homogenization; Upscaling Procedure; Effective Coefficient; Flow in Porous Media
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Applied Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
Warrick, Arthur W.
Committee Chair:
Warrick, Arthur W.

Full metadata record

DC FieldValue Language
dc.language.isoENen_US
dc.titleMultiscale Analytical Solutions and Homogenization of n-Dimensional Generalized Elliptic Equationsen_US
dc.creatorSviercoski, Rosangelaen_US
dc.contributor.authorSviercoski, Rosangelaen_US
dc.date.issued2005en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractIn this dissertation, we present multiscale analytical solutions, in the weak sense, to the generalized Laplace's equation in Ω ⊂ Rⁿ, subject to periodic and nonperiodic boundary conditions. They are called multiscale solutions since they depend on a coefficient which takes a wide possible range of scales. We define forms of nonseparable coefficient functions in Lᵖ(Ω) such that the solutions are valid for the periodic and nonperiodic cases. In the periodic case, one such solution corresponds to the auxiliary cell problem in homogenization theory. Based on the proposed analytical solution, we were able to write explicitly the analytical form for the upscaled equation with an effective coefficient, for linear and nonlinear cases including the one with body forces. This was done by performing the two-scale asymptotic expansion for linear and nonlinear equations in divergence form with periodic coefficient. We proved that the proposed homogenized coefficient satisfies the Voigt-Reiss inequality. By performing numerical experiments and error analyses, we were able to compare the heterogeneous equation and its homogenized approximation in order to define criteria in terms of allowable heterogeneity in the domain to obtain a good approximation. The results presented, in this dissertation, have laid mathematical groundwork to better understand and apply multiscale processes under a deterministic point of view.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectMultiscale Analytical Solutionsen_US
dc.subjectHomogenizationen_US
dc.subjectUpscaling Procedureen_US
dc.subjectEffective Coefficienten_US
dc.subjectFlow in Porous Mediaen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorWarrick, Arthur W.en_US
dc.contributor.chairWarrick, Arthur W.en_US
dc.contributor.committeememberBrio, Moyseyen_US
dc.contributor.committeememberNeuman, Shlomoen_US
dc.contributor.committeememberFerre, Paulen_US
dc.contributor.committeememberWinter, Larryen_US
dc.identifier.proquest1016en_US
dc.identifier.oclc71795023en_US
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