# American Institute of Mathematical Sciences

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

 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.
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 & Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279
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