DCDS-B

In this paper, we study a fully discrete finite element method with second order accuracy in time for the equations of motion arising in the Oldroyd model of viscoelastic fluids. This method is based on a finite element approximation for the space discretization and the Crank-Nicolson/Adams-Bashforth scheme for the time discretization. The integral term is discretized by the trapezoidal rule to match with the second order accuracy in time. It leads to a linear system with a constant matrix and thus greatly increases the computational efficiency. Taking the nonnegativity of the quadrature rule and the technique of variable substitution for the trapezoidal rule approximation, we prove that this fully discrete finite element method is almost unconditionally stable and convergent. Furthermore, by the negative norm technique, we derive the $H^1$ and $L^2$-optimal error estimates of the velocity and the pressure.

DCDS-B

In this article we compare the post-processing Galerkin (PPG) method
with the reformed PPG method of integrating the two-dimensional
Navier-Stokes equations in the case of non-smooth initial data
$u_0 \epsilon\in H^1_0(\Omega)^2$ with div$u_0=0$ and $f,~f_t\in
L^\infty(R^+;L^2(\Omega)^2)$. We give the global error estimates
with $H^1$ and $L^2$-norm
for these methods.
Moreover, if the data $\nu$
and the $\lim_{t \rightarrow \infty}f(t)$ satisfy the uniqueness condition,
the global error estimates with $H^1$ and $L^2$-norm are uniform in
time $t$. The difference between the PPG method and the reformed PPG
method is that their error bounds are of the same forms on the
interval $[1,\infty)$ and the reformed PPG method has a better error
bound than the PPG method on the interval $[0,1]$.

DCDS-B

In this article, we provide some asymptotic behaviors of linearized viscoelastic flows in a general two-dimensional domain with certain parameters small and the time variable large.

DCDS

In this paper, we present an Oseen coupling problem to approximate the two
dimensional exterior unsteady Navier-Stokes problem with the nonhomogeneous
boundary conditions. The Oseen coupling problem consists of the
Navier-Stokes equations in a bounded region and the Oseen equations in an
unbounded region. Then we derive the reduced Oseen coupling problem by use of
the integral representations of the solution of the Oseen equations in an
unbounded region. Moreover, we present the Galerkin approximation and the
nonlinear Galerkin approximation for the reduced Oseen coupling problem. By
analysing their convergence rates, we find that the nonlinear Galerkin
approximation provides the same convergence order as the classical Galerkin
approximation if we choose the space discrete parameter $H=O(h^{1/2})$.
However, in this approximation, the nonlinearity is treated on the coarse grid
finite element space and only the linear problem needs to be solved on the fine
grid finite element space.

DCDS-B

In this paper, we present the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. The Galerkin mixed finite element satisfying inf-sup condition is used for the spatial discretization and the temporal treatment is implicit/explict scheme, which is Euler implicit scheme for the linear terms and explicit scheme for the nonlinear term. We prove that this method is almost unconditionally convergent and obtain the optimal $H^1-L^2$ error estimate of the numerical velocity-pressure under the hypothesis of $H^2$-regularity of the solution for the three dimensional nonstationary Navier-Stokes equations. Finally some numerical experiments are carried out to demonstrate the effectiveness of the method.

DCDS

In this article, we provide the uniform $H^2$-regularity results with respect to $t$ of the solution and its time
derivatives for the 2D Cahn-Hilliard equation. Based on sharp a priori estimates for the solution of problem under the assumption on the initial value,
we show that the $H^2$-regularity
of the solution and its first and second order time derivatives only depend on
$\epsilon^{-1}$.

DCDS-B

In this article, we study the long time numerical stability and asymptotic behavior for the viscoelastic Oldroyd fluid motion equations. Firstly, with the Euler semi-implicit scheme for the temporal discretization, we deduce the global $H^2-$stability result for the fully discrete finite element solution. Secondly, based on the uniform stability of the numerical solution, we investigate the discrete asymptotic behavior and claim that the viscoelastic Oldroyd problem converges to the stationary Navier-Stokes flows if the body force $f(x,t)$ approaches to a steady-state $f_\infty(x)$ as $t\rightarrow\infty$. Finally, some numerical experiments are given to verify the theoretical predictions.

DCDS-B

In this article, a fully discrete finite element method is
considered for the viscoelastic fluid motion equations arising in
the two-dimensional Oldroyd model. A finite element method is
proposed for the spatial discretization and the time discretization
is based on the backward Euler scheme. Moreover, the stability and
optimal error estimates in the $L^2$- and $H^1$-norms for the
velocity and $L^2$-norm for the pressure are derived for all time
$t>0.$ Finally, some numerical experiments are shown to verify the
theoretical predictions.

CPAA

A reaction-diffusion system, based on the
cubic autocatalytic reaction scheme, with the prescribed
concentration boundary conditions is considered. The linear
stability of the unique spatially homogeneous steady state solution
is discussed in detail to reveal a necessary condition for the
bifurcation of this solution. The spatially non-uniform stationary
structures, especially bifurcating from the double eigenvalue, are
studied by the use of Lyapunov-Schmidt technique and singularity
theory. Further information about the multiplicity and stability of the bifurcation
solutions are obtained. Numerical examples are presented to support
our theoretical results.

DCDS

In this paper, the asymptotic analysis of the two-dimensional viscoelastic Oldroyd flows is presented. With the physical constant $\rho/\delta$ approaches zero, where $\rho$ is the viscoelastic coefficient and $1/\delta$ the relaxation time, the viscoelastic Oldroyd fluid motion equations converge to the viscous model known as the famous Navier-Stokes equations. Both the continuous and discrete uniform-in-time asymptotic errors are provided. Finally, the theoretical predictions are confirmed by some numerical experiments.