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

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  • 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.
    Mathematics Subject Classification: 35K55, 35B40, 65M12, 65M20.

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