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Title: Finite element approximation of a non-local problem in non-fickian polymer diffusion
Authors: Shaw, S
Keywords: A priori error estimates;Nonlinear diffusion;Non-Fickian diffusion;Finite element method;Linearisation;Extrapolation;Implicit Euler;Crank-Nicolson
Issue Date: 2011
Publisher: Institute for Scientific Computing and Information
Citation: International Journal of Numerical Analysis and Modeling, 8(2): 226 - 251, Jan 2011
Abstract: The problem of non-local nonlinear non-Fickian polymer diffusion as modelled by a diffusion equation with a nonlinearly coupled boundary value problem for a viscoelastic ‘pseudostress’ is considered (see, for example, DA Edwards in Z. angew. Math. Phys., 52, 2001, pp. 254—288). We present two numerical schemes using the implicit Euler method and also the Crank-Nicolson method. Each scheme uses a Galerkin finite element method for the spatial discretisation. Special attention is paid to linearising the discrete equations by extrapolating the value of the nonlinear terms from previous time steps. A priori error estimates are given, based on the usual assumptions that the exact solution possesses certain regularity properties, and numerical experiments are given to support these error estimates. We demonstrate by example that although both schemes converge at their optimal rates the Euler method may be more robust than the Crank-Nicolson method for problems of practical relevance.
Description: This is the post-print version of the Article. Copyright @ 2011 Institute for Scientific Computing and Information
ISSN: 1705-5105
Appears in Collections:Publications
Dept of Mathematics Research Papers
Mathematical Sciences

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