Artículo
Abstract:
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.
Registro:
| Documento: | Artículo |
| Título: | Numerical approximation of a parabolic problem with a nonlinear boundary condition |
| Autor: | Duran, R.G.; Etcheverry, J.I.; Rossi, J.D. |
| Filiación: | Departamento de Matemática, Fac. de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina |
| Palabras clave: | Blow up; Nonlinear boundary conditions; Numerical approximations |
| Año: | 1998 |
| Volumen: | 4 |
| Número: | 3 |
| Página de inicio: | 497 |
| Página de fin: | 506 |
| Título revista: | Discrete and Continuous Dynamical Systems |
| Título revista abreviado: | Discrete Contin. Dyn. Syst. |
| ISSN: | 10780947 |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10780947_v4_n3_p497_Duran |
Referencias:
- Hu, B., Yin, H.-M., The Profile Near Blow-up Time for the Solution of the Heat Equation with a Nonlinear Boundary Condition (1994) Trans. Amer. Math. Soc, 346 (1), pp. 117-135
- Ciarlet, P., (1978) The Finite Element Method for Elliptic Problems, , North Holland
- Douglas Jr., J., Dupont, T., Galerkin Methods for Parabolic Equations with Nonlinear Boundary Conditions (1973) Numer. Math., 20, pp. 213-237
- Levine, H.A., Payne, L.E., Nonexistence Theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time (1974) Jour. Diff. Eq., 16, pp. 319-334
- López Gómez, J., Márquez, V., Wolanski, N., Blow-up Results and Localization of Blow-up Points for the Heat Equation with a Nonlinear Boundary Condition (1991) Jour. Diff. Eq., 92 (2), pp. 384-401
- Luskin, M., A Galerkin Method for Nonlinear Parabolic Equations with Nonlinear Boundary Conditions (1979) SIAM J. Numer. Anal., 16 (2), pp. 284-299
- Rial, D.F., Rossi, J.D., Blow-up Results and Localization of Blow-up Points in an N-Dimensional Smooth Domain (1997) Duke Math. Jour, 88 (2), pp. 391-405
- Walter, W., On Existence and Nonexistence in the Large of Solutions of Parabolic Differential Equations with a Nonlinear Boundary Condition (1975) SIAM J. Math. Anal., 6 (1), pp. 85-90
- Wang, M., Wu, Y., Global existence and blow-up problems for quasilinear parabolic equations with nonlinear boundary conditions (1993) SIAM J. Math. Anal., 24 (6), pp. 1515-1521
Citas:
---------- APA ----------
Duran, R.G., Etcheverry, J.I. & Rossi, J.D. (1998) . Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 4(3), 497-506.Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10780947_v4_n3_p497_Duran [ ]
---------- CHICAGO ----------
Duran, R.G., Etcheverry, J.I., Rossi, J.D. "Numerical approximation of a parabolic problem with a nonlinear boundary condition" . Discrete and Continuous Dynamical Systems 4, no. 3 (1998) : 497-506.Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10780947_v4_n3_p497_Duran [ ]
---------- MLA ----------
Duran, R.G., Etcheverry, J.I., Rossi, J.D. "Numerical approximation of a parabolic problem with a nonlinear boundary condition" . Discrete and Continuous Dynamical Systems, vol. 4, no. 3, 1998, pp. 497-506.Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10780947_v4_n3_p497_Duran [ ]
---------- VANCOUVER ----------
Duran, R.G., Etcheverry, J.I., Rossi, J.D. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete Contin. Dyn. Syst. 1998;4(3):497-506.Available from: https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10780947_v4_n3_p497_Duran [ ]
