Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model

On a Linearized Backward Euler Method for the Equations of Motion of Oldroyd Fluids of Order One

“…Note that the L 2 -estimates of I n 2.2 and I n 2,3 in (3.13) follow easily as We repeat the analysis for estimating A 1/2 I n 2,3 in Theorem 1, but now e i+1 is made free of A 1/2 . Thus, we obtain results obtained in [26]. Since the problem (1.1)-(1.3) can be thought of as an integral perturbation of the Navier-Stokes equations, we would like to investigate how far the results on finite element analysis combined with higher order time discretizations of the Navier-Stokes equations [15], [16], [22] can be carried over to the present case.…”

“…While Akhmatov and Oskolkov [2] applied a finite difference scheme to the equation of motion arising in the Oldroyd model, Cannon et al [5] analyzed a modified nonlinear Galerkin scheme for a periodic problem using spectral Galerkin procedure and discussed the rates of convergence for the semidiscrete approximations. Recently, Pani and Yuan [26] and He et al [12] applied finite element methods to discretize the spatial variables and derived optimal error estimates for the problems (1.1)-(1.3) without using nonlocal compatibility conditions. In all these pappers [5], [26], [12], only semidiscrete approximations are discussed keeping the time variable continuous.…”

“…Recently, Pani and Yuan [26] and He et al [12] applied finite element methods to discretize the spatial variables and derived optimal error estimates for the problems (1.1)-(1.3) without using nonlocal compatibility conditions. In all these pappers [5], [26], [12], only semidiscrete approximations are discussed keeping the time variable continuous. In this article, we have proposed and analyzed a time discretization scheme based on linearized modification of the backward Euler Downloaded 12/30/12 to 152.3.102.242.…”

On error estimates of the fully discrete penalty method for the viscoelastic flow problem

“…For the time semi-discrete, Pani et al investigated a linearized backward Euler method in [27]. For the spatial semi-discrete, the reader is referred to Akhmatov and Oskolkov [4] for the difference schemes, and He et al [16] Pani and Yuan [26] for the conforming finite element method, and Cannon et al [9] for a modified nonlinear Galerkin method. Recently, Wang et al [34] extended the analysis to a fully discrete finite element scheme.…”

Optimal error estimates of the penalty method for the linearized viscoelastic flows

“…These papers dealt with the equations of existence, uniqueness and continuous dependence of the solution upon the data. For the numerical approximations to the problems (1) and (2), for the spatial discretization, we refer to Akhmatov and Oskolkov [4] for difference schemes, Cannon et al [7] for a modified nonlinear Galerkin method, He et al [12] and Pani and Yuan [21] for the conforming finite element method and Pani et al [22] for a linearized backward Euler method for the time discretization. The linearized problem of Equations (1) and (2) is also considered by He and Li [11] for the asymptotic behaviour.…”