Search

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]

Simulations of blood flows in arteries require numerical solutions of fluidstructure interactions involving Navier-Stokes equations coupled with large displacement visco-elasticity for the vessels. Among the various simplifications which have been proposed, the surface pressure model leads to a hierarchy of simpler models including one that involves only the pressure. The model exhibits fundamental frequencies which can be computed and compared with the pulse. Yet unconditionally stable time discretizations can be constructed by combining implicit time schemes with Galerkin-characteristic discretization of the convection terms in the Navier-Stokes equations. Such problems with prescribed pressure on the walls will be shown to be efficient and accurate as an approximation of the full fluid structure interaction problem. [ABSTRACT FROM AUTHOR]

A splitting positive definite mixed finite element method is proposed for second-order viscoelasticity wave equation. The proposed procedure can be split into three independent symmetric positive definite integro-differential sub-system and does not need to solve a coupled system of equations. Error estimates are derived for both semidiscrete and fully discrete schemes. The existence and uniqueness for semidiscrete scheme are proved. Finally, a numerical example is provided to illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]