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Finite element approximation of a non-local problem in non-Fickian polymer diffusion. (English) Zbl 1208.82086

Summary: 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, D. A. Edwards, Z. Angew. Math. Phys. 52, No. 2, 254–288 (2001; Zbl 1160.35328)]. 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.

MSC:

82D60 Statistical mechanics of polymers
74S05 Finite element methods applied to problems in solid mechanics
74S20 Finite difference methods applied to problems in solid mechanics
76R50 Diffusion
74D10 Nonlinear constitutive equations for materials with memory

Citations:

Zbl 1160.35328

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