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IMA Journal of Numerical Analysis Advance Access published online on March 16, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn071
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Finite-element approximation of non-Fickian polymer diffusion

Norbert Bauermeister and Simon Shaw{dagger}

Brunel Institute of Computational Mathematics, Brunel University, Uxbridge UB8 3PH, UK

{dagger} Corresponding author. Email: simon.shaw{at}brunel.ac.uk

Received on 11 January 2008. Revised on 24 July 2008.


  Abstract

The problem of nonlinear non-Fickian polymer diffusion as modelled by a diffusion equation with an adjoined spatially local evolution equation for a viscoelastic stress is considered. We present numerical schemes based, spatially, on the Galerkin finite-element method and, temporally, on the Crank–Nicolson method. Special attention is paid to linearizing the discrete equations by extrapolating the value of the nonlinear term from previous time steps. Optimal a priori error estimates are given, based on the assumption that the exact solution possesses certain regularity properties, and numerical experiments are given to support these error estimates.

Key Words: a priori error estimates; nonlinear diffusion; non-Fickian diffusion; finite-element method


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