Discontinuous Galerkin Finite Element Approximation of Nonlinear Non-Fickian Diffusion in Viscoelastic Polymers

We consider discrete schemes for a nonlinear model of non-Fickian diffusion in viscoelastic polymers. The model is motivated by, but not the same as, that proposed by Cohen et al. in SIAM J. Appl. Math., 55, pp. 348–368, 1995. The spatial discretisation is effected with both the symmetric and non-symmetric interior penalty discontinuous Galerkin finite element method, and the time discretisation is of Crank-Nicolson type. We also discuss two means of handling the nonlinearity: either implicitly, which requires the solution of nonlinear equations at each time level, or through a linearisation based on extrapolating from previous time levels. The same optimal orders of convergence are proven in both cases and, to verify this, some numerical results are also given for the linearised scheme.