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May  2002, 2(2): 279-294. doi: 10.3934/dcdsb.2002.2.279

Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions

G. Acosta 1, , Julián Fernández Bonder 2, , P. Groisman 1, and  J.D. Rossi 1,

1. 

Department of Mathematics, FCEyN, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina, Argentina, Argentina
2. 

Departamento de Matematica, FCEyN, UBA, 1428 Buenos Aires, Argentina

Received  December 2001 Published  February 2002

In this paper we study the asymptotic behavior of a semidiscrete numerical approximation for the heat equation, $u_t = \Delta u$, in a bounded smooth domain with a nonlinear flux boundary condition, $(\partial u)/(\partial\eta)= u^p$. We focus in the behavior of blowing up solutions. We prove that every numerical solution blows up in finite time if and only if $p > 1$ and that the numerical blow-up time converges to the continuous one as the mesh parameter goes to zero. Also we show that the blow-up rate for the numerical scheme is different from the continuous one. Nevertheless we find that the blow-up set for the numerical approximations is contained in a small neighborhood of the blow-up set of the continuous problem when the mesh parameter is small enough.
Keywords: parabolic equations, asymptotic behaviour., semidiscretization in space, Blow-up.
Mathematics Subject Classification: 35K55, 35B40, 65M12, 65M2.
Citation: G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279
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