Numerical approximation of a parabolic problem with a nonlinear boundary condition
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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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_10780947_v4_n3_p497_Duran |
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todo:paper_10780947_v4_n3_p497_Duran2023-10-03T16:03:41Z Numerical approximation of a parabolic problem with a nonlinear boundary condition Duran, R.G. Etcheverry, J.I. Rossi, J.D. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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, R.G. Etcheverry, J.I. Rossi, J.D. 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. |
| format |
JOUR |
| author |
Duran, R.G. Etcheverry, J.I. Rossi, J.D. |
| author_facet |
Duran, R.G. Etcheverry, J.I. Rossi, J.D. |
| author_sort |
Duran, R.G. |
| 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 |
| url |
http://hdl.handle.net/20.500.12110/paper_10780947_v4_n3_p497_Duran |
| work_keys_str_mv |
AT duranrg numericalapproximationofaparabolicproblemwithanonlinearboundarycondition AT etcheverryji numericalapproximationofaparabolicproblemwithanonlinearboundarycondition AT rossijd numericalapproximationofaparabolicproblemwithanonlinearboundarycondition |
| _version_ |
1807319314721669120 |