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

Béatrice Rivière and Simon Shaw
SIAM Journal on Numerical Analysis
Vol. 44, No. 6 (2006), pp. 2650-2670
Stable URL: http://www.jstor.org/stable/40232911
Page Count: 21
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Discontinuous Galerkin Finite Element Approximation of Nonlinear Non-Fickian Diffusion in Viscoelastic Polymers
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Abstract

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, White, and Witelski in SIAM J. Appl. Math., 55 (1995), pp. 348-368. The spatial discretization is effected with both the symmetric and nonsymmetric interior penalty discontinuous Galerkin finite element method, and the time discretization 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 linearization 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 linearized scheme.

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