Numerical Methods (Finite Element) for Time-Dependent Partial Differential Equations
This paper surveys reasons why the Ritz method and the Galerkin method are not efficient and why these methods can not be applied directly, for time dependent problems. It also introduces methods that are used for those problems. For a linear boundary value problem defined by a positive definite sym...
Main Author: | |
---|---|
Format: | Others |
Published: |
DigitalCommons@USU
1982
|
Subjects: | |
Online Access: | https://digitalcommons.usu.edu/etd/6978 https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=8083&context=etd |
id |
ndltd-UTAHS-oai-digitalcommons.usu.edu-etd-8083 |
---|---|
record_format |
oai_dc |
spelling |
ndltd-UTAHS-oai-digitalcommons.usu.edu-etd-80832019-10-13T06:02:40Z Numerical Methods (Finite Element) for Time-Dependent Partial Differential Equations Watanabe, Masaji This paper surveys reasons why the Ritz method and the Galerkin method are not efficient and why these methods can not be applied directly, for time dependent problems. It also introduces methods that are used for those problems. For a linear boundary value problem defined by a positive definite symmetric (self-adjoint) operator, the existence and the convergence of the Ritz approximation are guaranteed. In non-symmetric case, Lax-Milgram lemma assures the existence and the convergence of the Galerkin approximation for H1/2(Ω)-elliptic operator. Since time dependent problems are hyperbolic or parabolic, the existence and the convergence of approximations by those methods are not guaranteed. Moreover, those methods were originally developed for boundary value problems. Thus new techniques are introduced in order to extend those methods to initial-boundary value problems. 1982-05-01T07:00:00Z text application/pdf https://digitalcommons.usu.edu/etd/6978 https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=8083&context=etd Copyright for this work is held by the author. Transmission or reproduction of materials protected by copyright beyond that allowed by fair use requires the written permission of the copyright owners. Works not in the public domain cannot be commercially exploited without permission of the copyright owner. Responsibility for any use rests exclusively with the user. For more information contact digitalcommons@usu.edu. All Graduate Theses and Dissertations DigitalCommons@USU Mathematics |
collection |
NDLTD |
format |
Others
|
sources |
NDLTD |
topic |
Mathematics |
spellingShingle |
Mathematics Watanabe, Masaji Numerical Methods (Finite Element) for Time-Dependent Partial Differential Equations |
description |
This paper surveys reasons why the Ritz method and the Galerkin method are not efficient and why these methods can not be applied directly, for time dependent problems. It also introduces methods that are used for those problems. For a linear boundary value problem defined by a positive definite symmetric (self-adjoint) operator, the existence and the convergence of the Ritz approximation are guaranteed. In non-symmetric case, Lax-Milgram lemma assures the existence and the convergence of the Galerkin approximation for H1/2(Ω)-elliptic operator. Since time dependent problems are hyperbolic or parabolic, the existence and the convergence of approximations by those methods are not guaranteed. Moreover, those methods were originally developed for boundary value problems. Thus new techniques are introduced in order to extend those methods to initial-boundary value problems. |
author |
Watanabe, Masaji |
author_facet |
Watanabe, Masaji |
author_sort |
Watanabe, Masaji |
title |
Numerical Methods (Finite Element) for Time-Dependent Partial Differential Equations |
title_short |
Numerical Methods (Finite Element) for Time-Dependent Partial Differential Equations |
title_full |
Numerical Methods (Finite Element) for Time-Dependent Partial Differential Equations |
title_fullStr |
Numerical Methods (Finite Element) for Time-Dependent Partial Differential Equations |
title_full_unstemmed |
Numerical Methods (Finite Element) for Time-Dependent Partial Differential Equations |
title_sort |
numerical methods (finite element) for time-dependent partial differential equations |
publisher |
DigitalCommons@USU |
publishDate |
1982 |
url |
https://digitalcommons.usu.edu/etd/6978 https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=8083&context=etd |
work_keys_str_mv |
AT watanabemasaji numericalmethodsfiniteelementfortimedependentpartialdifferentialequations |
_version_ |
1719267862132031488 |