Finite element approximation of a non-local problem in non-Fickian polymer diffusion
S. Shaw 11 Brunel Institute of Computational Mathematics, Brunel University UB8 3PH, England.
Received by the editors April 30, 2010 and, in revised form, July 22, 2010.
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.AMS subject classifications: 74S05, 74S20, 76R50, 74D10, 82D60
Key words: a priori error estimates, nonlinear diffusion, non-Fickian diffusion, finite element method, linearisation, extrapolation, implicit Euler, Crank-Nicolson.