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The stability and convergence of a linearly extrapolated second order backward difference (BDF2-LE) time-stepping scheme for solving viscoelastic fluid flow in $\mathbb{R}^{d}$ , $d=2,3$ , are presented in this paper. The time discretization is based on the implicit scheme for the linear term and the two-step linearly extrapolated scheme for the nonlinear term. Mixed finite element (MFE) method is applied for the spatial discretization. The approximations of stress tensor σ, velocity vector u and pressure p are $P_{m}$ -discontinuous, $P_{k}$ -continuous and $P_{q}$ -continuous elements, respectively. Upwinding needed for convection of σ is made by a discontinuous Galerkin (DG) FE method. For the time step △ t small enough, the existence of an approximate solution is proven. If $m, k \geqslant \frac{d}{2}$ , $q+1\geqslant\frac{d}{2}$ , and $\triangle t \leqslant C_{0} h^{\frac{d}{4}}$ , then the discrete $H^{1}$ and $L^{2}$ errors for the velocity and stress, and $L^{2}$ error for the pressure, are bounded by $C(\triangle t^{2}+h^{\min\{m,k,q+1\}})$ , where h denotes the mesh size. The derived theoretical results are supported by numerical tests. [ABSTRACT FROM AUTHOR]

We propose mixed finite element methods for the standard linear solid model in viscoelasticity and prove a priori error estimates. In our mixed formulation the governing equations of the problem become a symmetric hyperbolic system, so we can use standard techniques for a priori error estimates and time discretization. Numerical results illustrating our theoretical analysis are included. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a fully discrete local discontinuous Galerkin approximation of a non-linear non-Fickian diffusion model in viscoelastic polymers. For the spatial discretization, we adopt local discontinuous Galerkin finite element method and for the time discretization we use backward Euler method. We derive the stability estimate and a priori error estimate for the discrete scheme. Numerical examples are given to verify the theoretical findings. [ABSTRACT FROM PUBLISHER]