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IMA Journal of Numerical Analysis 2003 23(3):439-463; doi:10.1093/imanum/23.3.439
© 2003 by Institute of Mathematics and its Applications
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Adaptive numerical schemes for a parabolic problem with blow-up

Raúl Ferreira1, Pablo Groisman2 and Julio D. Rossi2

1 Depto. de Matemáticas, U. Autónoma de Madrid, 28049 Madrid, Spain 2 Depto. de Matemática, FCEyN., UBA, (1428), Buenos Aires, Argentina

In this paper we present adaptive procedures for the numerical study of positive solutions of the following problem:

ut = uxx (x, t) (0, 1) x [0, T),

ux(0, t) = 0 t [0, T),

ux(1, t) = up(1, t) t [0, T),

u(x, 0) = u0(x) x (0, 1),

with p > 1. We describe two methods. The first one refines the mesh in the region where the solution becomes bigger in a precise way that allows us to recover the blow-up rate and the blow-up set of the continuous problem. The second one combines the ideas used in the first one with moving mesh methods and moves the last points when necessary. This scheme also recovers the blow-up rate and set. Finally, we present numerical experiments to illustrate the behaviour of both methods.

Key Words: numerical blow-up; heat equation; nonlinear boundary conditions; adaptive mesh

Received 8 April 2002. Revised 23 October 2002.


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