DCDS
On uniform in time $H^2$-regularity of the solution for the 2D Cahn-Hilliard equation
Xinlong Feng Yinnian He
Discrete & Continuous Dynamical Systems - A 2016, 36(10): 5387-5400 doi: 10.3934/dcds.2016037
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 $\epsilon^{-1}$.
keywords: sharp a priori estimates mixed weak formulation Cahn-Hilliard equation time derivatives of solution. $H^2$-regularity
DCDS-B
Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one
Yingwen Guo Yinnian He
Discrete & Continuous Dynamical Systems - B 2015, 20(8): 2583-2609 doi: 10.3934/dcdsb.2015.20.2583
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.
keywords: mixed finite element Adams-Bashforth scheme Crank-Nicolson scheme. Viscoelastic fluids Oldroyd fluids of order one
DCDS-B
Reformed post-processing Galerkin method for the Navier-Stokes equations
Yinnian He R. M.M. Mattheij
Discrete & Continuous Dynamical Systems - B 2007, 8(2): 369-387 doi: 10.3934/dcdsb.2007.8.369
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 $[0,1]$.
keywords: Galerkin method Post-processing Galerkin method Error estimate. Navier-Stokes equations
DCDS-B
Asymptotic behavior of linearized viscoelastic flow problem
Yinnian He Yi Li
Discrete & Continuous Dynamical Systems - B 2008, 10(4): 843-856 doi: 10.3934/dcdsb.2008.10.843
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.
keywords: Viscoelastic flows; Navier-Stokes flows; Euler flows; Asymptotic behavior.
DCDS
Nonlinear Galerkin approximation of the two dimensional exterior Navier-Stokes problem
Yinnian He Kaitai Li
Discrete & Continuous Dynamical Systems - A 1996, 2(4): 467-482 doi: 10.3934/dcds.1996.2.467
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.
keywords: Navier-Stokes equations nonlinear Galerkin approximation. finite element oseen equations exterior domain
DCDS-B
The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations
Jian Su Yinnian He
Discrete & Continuous Dynamical Systems - B 2017, 22(9): 3421-3438 doi: 10.3934/dcdsb.2017173

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.

keywords: Navier-Stokes equations mixed finite element Euler implicit/explicit scheme convergence error estimate
DCDS-B
Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows
Kun Wang Yinnian He Yanping Lin
Discrete & Continuous Dynamical Systems - B 2012, 17(5): 1551-1573 doi: 10.3934/dcdsb.2012.17.1551
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.
keywords: Oldroyd model Viscoelastic flows asymptotic analysis. $H^2-$stability long time behavior
DCDS-B
Fully discrete finite element method for the viscoelastic fluid motion equations
Kun Wang Yinnian He Yueqiang Shang
Discrete & Continuous Dynamical Systems - B 2010, 13(3): 665-684 doi: 10.3934/dcdsb.2010.13.665
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.
keywords: time discretization finite element method Viscoelastic fluid motion equations long-time error estimate. Oldroyd model
CPAA
Steady-state solutions and stability for a cubic autocatalysis model
Mei-hua Wei Jianhua Wu Yinnian He
Communications on Pure & Applied Analysis 2015, 14(3): 1147-1167 doi: 10.3934/cpaa.2015.14.1147
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.
keywords: singularity theory. steady state solutions Cubic autocatalysis model dynamical stability Lyapunov-Schmidt procedure
DCDS
Asymptotic analysis of the equations of motion for viscoelastic oldroyd fluid
Kun Wang Yangping Lin Yinnian He
Discrete & Continuous Dynamical Systems - A 2012, 32(2): 657-677 doi: 10.3934/dcds.2012.32.657
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.
keywords: Viscoelastic flows long time behavior Oldroyd model asymptotic behavior Navier-Stokes equations.

Year of publication

Related Authors

Related Keywords

[Back to Top]