Numerical approximation of a parabolic problem with a nonlinear boundary condition

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We analyze numerical approximations of positive solutions of a heat equation with a nonlinear flux condition which produces blow up of the solutions. By a semidiscretization using finite elements in the space variable we obtain a system of ordinary differential equations which is expected to be an a...

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Detalles Bibliográficos
Autores principales:	Duran, Ricardo Guillermo, Etcheverry, Javier Ignacio, Rossi, Julio Daniel
Publicado:	1998
Materias:	Blow up Nonlinear boundary conditions Numerical approximations
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10780947_v4_n3_p497_Duran http://hdl.handle.net/20.500.12110/paper_10780947_v4_n3_p497_Duran
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_10780947_v4_n3_p497_Duran
record_format	dspace
spelling	paper:paper_10780947_v4_n3_p497_Duran2023-06-08T16:05:36Z Numerical approximation of a parabolic problem with a nonlinear boundary condition Duran, Ricardo Guillermo Etcheverry, Javier Ignacio Rossi, Julio Daniel Blow up Nonlinear boundary conditions Numerical approximations We analyze numerical approximations of positive solutions of a heat equation with a nonlinear flux condition which produces blow up of the solutions. By a semidiscretization using finite elements in the space variable we obtain a system of ordinary differential equations which is expected to be an approximation of the original problem. Our objective is to analyze whether this system has a similar behaviour than the original problem. We find a necessary and sufficient condition for blow up of this system. However, this condition is slightly different than the one known for the original problem, in particular, there are cases in which the continuous problem has blow up while its semidiscrete approximation does not. Under certain assumptions we also prove that the numerical blow up time converges to the real blow-up time when the meshsize goes to zero. Our proofs are given in one space dimension. Similar arguments could be applied for higher dimensions but a further analysis is required. Fil:Duran, R.G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Etcheverry, J.I. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 1998 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10780947_v4_n3_p497_Duran http://hdl.handle.net/20.500.12110/paper_10780947_v4_n3_p497_Duran
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Blow up Nonlinear boundary conditions Numerical approximations
spellingShingle	Blow up Nonlinear boundary conditions Numerical approximations Duran, Ricardo Guillermo Etcheverry, Javier Ignacio Rossi, Julio Daniel Numerical approximation of a parabolic problem with a nonlinear boundary condition
topic_facet	Blow up Nonlinear boundary conditions Numerical approximations
description	We analyze numerical approximations of positive solutions of a heat equation with a nonlinear flux condition which produces blow up of the solutions. By a semidiscretization using finite elements in the space variable we obtain a system of ordinary differential equations which is expected to be an approximation of the original problem. Our objective is to analyze whether this system has a similar behaviour than the original problem. We find a necessary and sufficient condition for blow up of this system. However, this condition is slightly different than the one known for the original problem, in particular, there are cases in which the continuous problem has blow up while its semidiscrete approximation does not. Under certain assumptions we also prove that the numerical blow up time converges to the real blow-up time when the meshsize goes to zero. Our proofs are given in one space dimension. Similar arguments could be applied for higher dimensions but a further analysis is required.
author	Duran, Ricardo Guillermo Etcheverry, Javier Ignacio Rossi, Julio Daniel
author_facet	Duran, Ricardo Guillermo Etcheverry, Javier Ignacio Rossi, Julio Daniel
author_sort	Duran, Ricardo Guillermo
title	Numerical approximation of a parabolic problem with a nonlinear boundary condition
title_short	Numerical approximation of a parabolic problem with a nonlinear boundary condition
title_full	Numerical approximation of a parabolic problem with a nonlinear boundary condition
title_fullStr	Numerical approximation of a parabolic problem with a nonlinear boundary condition
title_full_unstemmed	Numerical approximation of a parabolic problem with a nonlinear boundary condition
title_sort	numerical approximation of a parabolic problem with a nonlinear boundary condition
publishDate	1998
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10780947_v4_n3_p497_Duran http://hdl.handle.net/20.500.12110/paper_10780947_v4_n3_p497_Duran
work_keys_str_mv	AT duranricardoguillermo numericalapproximationofaparabolicproblemwithanonlinearboundarycondition AT etcheverryjavierignacio numericalapproximationofaparabolicproblemwithanonlinearboundarycondition AT rossijuliodaniel numericalapproximationofaparabolicproblemwithanonlinearboundarycondition
_version_	1768546455844290560

Numerical approximation of a parabolic problem with a nonlinear boundary condition

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