TY - JOUR
T1 - Finite Element Approximation of a Non-Local Problem in Non-Fickian Polymer Diffusion
AU - S. Shaw
JO - International Journal of Numerical Analysis and Modeling
VL - 2
SP - 226
EP - 251
PY - 2011
DA - 2011/08
SN - 8
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/684.html
KW - a priori error estimates, nonlinear diffusion, non-Fickian diffusion, finite element method, linearisation, extrapolation, implicit Euler, Crank-Nicolson.
AB - <p style="text-align: justify;">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 &#39;pseudostress&#39;
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.</p>