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Discretizations of nonlinear dissipative evolution equations. Order and convergence.

Hansen, Eskil LU (2005)
Abstract
The theme of this thesis is to study discretizations of nonlinear dissipative evolution equations, which arise in e.g. advection-diffusion-reaction processes. The convergence analysis is conducted by first considering an abstract time discretization of the problem, which enables a decoupling of the time and spatial approximations, and secondly by introducing the spatial discretization as an evolution on a finite dimensional space.



For A-stable multistep methods and algebraically stable Runge-Kutta methods the very same global error bounds are obtained in this infinite dimensional setting as derived for stiff ODEs. Error bounds are also presented for full discretizations based on spatial Galerkin approximations.

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The theme of this thesis is to study discretizations of nonlinear dissipative evolution equations, which arise in e.g. advection-diffusion-reaction processes. The convergence analysis is conducted by first considering an abstract time discretization of the problem, which enables a decoupling of the time and spatial approximations, and secondly by introducing the spatial discretization as an evolution on a finite dimensional space.



For A-stable multistep methods and algebraically stable Runge-Kutta methods the very same global error bounds are obtained in this infinite dimensional setting as derived for stiff ODEs. Error bounds are also presented for full discretizations based on spatial Galerkin approximations.



In contrast to earlier studies, our analysis is not relying on linearization procedures, but on the fully nonlinear framework of logarithmic Lipschitz constants and a generalization of the classical B-convergence theory. (Less)
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author
supervisor
opponent
  • Professor Ostermann, Alexander, Leopold-Franzens-Universität, Innsbruck, Austria
organization
publishing date
type
Thesis
publication status
published
subject
keywords
kontroll, systems, numerisk analys, control, Datalogi, numerical analysis, Computer science, B-convergence, Dissipative maps, Logarithmic Lipschitz constants, Galerkin methods, Nonlinear evolution equations, Time discretizations, system
pages
84 pages
publisher
Numerical Analysis, Lund University
defense location
Room MH:C, Centre for Mathematical Sciences, Sölvegatan 18, Lund Institute of Technology
defense date
2005-12-09 13:15
external identifiers
  • other:ISRN: LUTFNA-1001-2005
ISSN
1404-0034
ISBN
91-628-6668-0
language
English
LU publication?
yes
id
d1cd9ba8-5807-4813-a741-3f6ac9b403ac (old id 545744)
date added to LUP
2007-09-27 15:42:33
date last changed
2017-02-13 13:11:26
@phdthesis{d1cd9ba8-5807-4813-a741-3f6ac9b403ac,
  abstract     = {The theme of this thesis is to study discretizations of nonlinear dissipative evolution equations, which arise in e.g. advection-diffusion-reaction processes. The convergence analysis is conducted by first considering an abstract time discretization of the problem, which enables a decoupling of the time and spatial approximations, and secondly by introducing the spatial discretization as an evolution on a finite dimensional space.<br/><br>
<br/><br>
For A-stable multistep methods and algebraically stable Runge-Kutta methods the very same global error bounds are obtained in this infinite dimensional setting as derived for stiff ODEs. Error bounds are also presented for full discretizations based on spatial Galerkin approximations.<br/><br>
<br/><br>
In contrast to earlier studies, our analysis is not relying on linearization procedures, but on the fully nonlinear framework of logarithmic Lipschitz constants and a generalization of the classical B-convergence theory.},
  author       = {Hansen, Eskil},
  isbn         = {91-628-6668-0},
  issn         = {1404-0034},
  keyword      = {kontroll,systems,numerisk analys,control,Datalogi,numerical analysis,Computer science,B-convergence,Dissipative maps,Logarithmic Lipschitz constants,Galerkin methods,Nonlinear evolution equations,Time discretizations,system},
  language     = {eng},
  pages        = {84},
  publisher    = {Numerical Analysis, Lund University},
  school       = {Lund University},
  title        = {Discretizations of nonlinear dissipative evolution equations. Order and convergence.},
  year         = {2005},
}