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IMA Journal of Numerical Analysis Advance Access originally published online on June 30, 2008
IMA Journal of Numerical Analysis 2009 29(3):690-711; doi:10.1093/imanum/drn030
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Attractors of set-valued partial differential equations under discretization

P. E. Kloeden{dagger}

Institut für Mathematik, Johann Wolfgang Goethe Universität, D-60054 Frankfurt am Main, Germany

J. Valero{ddagger}

Centro de Investigation Operativa, Universidad Miguel Hernández, Avenida de la Universidad s/n, ES-03202 Elche, Spain

{dagger} Corresponding author. Email: kloeden{at}math.uni-frankfurt.de

{ddagger} Email: jvalero{at}umh.es

Received on 11 May 2007. Revised on 27 February 2008.


  Abstract

The approximation of the global attractor of a dissipative set-valued reaction–diffusion equation is investigated when a Galerkin approximation is used to obtain a finite-dimensional inclusion equation, to which the linear implicit Euler scheme is then applied. The existence and upper semicontinuous convergence of the various attractors with decreasing time step and increasing dimension are established. The equivalence of the attractors with those of the corresponding convexified systems is also shown.

Key Words: set-valued reaction–diffusion equation; set-valued partial differential equation; Galerkin approximation; linear implicit Euler scheme; set-valued dynamical systems; global attractors; convexification


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