电报方程的时间间断时空有限元方法

为同时高精度逼近速度和位移,利用时间间断的时空有限元与降阶的思想,对一类电报方程的初边值问题建立一种时间间断时空有限元格式.利用有限差分方法与有限元方法相结合的技巧,证明了格式的稳定性和收敛性,得到了速度的L∞(L2)模和位移的L∞(H1)模最优误差估计.最后用数值算例验证了理论分析结果和所提算法的有效性.     

对流反应扩散方程的稳定化时间间断时空有限元解的误差估计

在流线迎风Petrov-Galerkin（SUPG）稳定化有限元数值格式的基础上,结合时间方向的变分离散,构造对流反应扩散方程的稳定化时间间断时空有限元格式.该类格式在工程上有一些数值模拟应用,但相关文献没有看到类似数值格式的理论证明.本文以Radau点为节点,构造时间方向的Lagrange插值多项式,证明了稳定化有限元解的稳定性,时间最大模、空间L²（Ω）-模误差估计.文中利用插值多项式和有限元方法相结合的技巧,解耦时空变量,去掉了时空网格的限制条件,提供了时间间断稳定化时空有限元方法的理论证明思路,克服了因时空变量统一导致的实际计算时的复杂性.     

非线性分数阶反应扩散方程组的间断时空有限元方法

利用时间间断空间连续的时空有限元方法构造了空间分数阶反应扩散方程组的可以逐时间层求解的全离散格式.在时间离散区间上,采用Radau积分公式,将插值理论与有限元理论相结合,给出了全离散格式解的存在唯一性结果,并证明了所给格式是无条件稳定的,进而详细给出最优阶L~∞(L~2)模误差估计过程.最后用数值算例验证了理论分析的正确性.     

一类三维变动区域上的非线性抛物型方程之全离散有限元形式

1 引言 1986年,L.Cermak和M.Zlamal研究了半导体器件中杂质的重新分布,对具有活动边界的二维非线性扩散问题。给出在时间方向上是一阶精度的全离散有限元格式。证明了格式最优的H~1模和次最优的L~2模估计。1989年.P.Lesaint和R.Touzani对一维变动区域上的热传导方程。经过坐标变换,给出了在固定区域上的全离散有限元格式和最优的L~2模估计。1990年,梁国平和陈志明利用时空有限元,给出了变动区域上线性抛物型的方程的全离散变网格有限元格式。证明了最优的L~2收敛性。本文考虑了一类具有活动边界的三维     

非线性sine-Gordon方程的双线性元分析及外推

研究双线性元对一类非线性sine-Gordon方程的有限元逼近.利用该元的高精度结果和对时间t的导数转移技巧,得到了H~1模意义下的超逼近性.进一步地,通过运用插值后处理技术,给出了H~1模意义下的超收敛结果.与此同时,通过构造一个新的外推格式,导出了与线性问题情形相同的三阶外推解.最后给出了一种全离散逼近格式下的最优误差估计.     

电报方程的类Wilson非协调有限元分析

本文在矩形网格上讨论了半离散和全离散格式下电报方程的类Wilson非协调有限元逼近.利用该元在H1模意义下O(h2)阶的相容误差结果,平均值理论和关于时间t的导数转移技巧得到了超逼近性.进而,借助于插值后处理方法导出了超收敛结果.又由于该元在H1模意义下的相容误差可以达到O(h3)阶,构造了新的外推格式,给出了比传统误差估计高两阶的外推估计.最后,对于给出的全离散逼近格式得到了最优误差估计.     

Sobolev方程的一类低阶非协调元高精度全离散格式分析

本文针对Sobolev方程提出一类低阶非协调有限元全离散格式,对时间变量具有二阶精度,对空间变量得到能量模意义下的超逼近和全局超收敛结果.最后给出的数值算例验证了理论分析的正确性.     

抛物型积分微分方程新混合元格式的超逼近分析

基于双二次元及其梯度空间,建立了抛物型积分微分方程的一种新混合有限元逼近格式.在不需要Ritz-Volterra投影的前提下,直接利用双二次元插值的高精度结果及关于时间变量的导数转移技巧,在半离散格式下,得到了原始变量u和中间变量p=▽u+integral from n=0 to t▽u(s)ds分别关于H~1模和L~2模的O(h~4)阶超逼近结果,相比插值误差估计,提高了二阶精度.与此同时,对向后Euler格式,导出了u和p分别在H~1模与L~2模意义下的O(h~4+τ)阶超逼近;对Crank-Nicolson-Galerkin格式,在L~2模意义下证明了u和p分别具有O(h~4+τ~2)和O(h~3+τ~2)阶的超逼近性质.其中,h,τ分别表示空间剖分参数和时间步长,t代表时间变量.     

抛物型方程一个新的非协调混合元超收敛性分析及外推

本文研究了抛物型方程在新混合元格式下的非协调混合有限元方法. 在抛弃传统有限元分析的必要工具-Ritz 投影算子的前提下,直接利用单元的插值性质,运用高精度分析和对时间t的导数转移技巧,借助于插值后处理技术,分别导出了关于原始变量u的H¹-模和通量p=▽u在L²-模下的O（h²）阶超逼近性质和整体超收敛. 进一步,通过构造合适的辅助问题,运用Richardson 外推格式,得到了具有更高精度O（h³）阶的外推结果. 最后,给出了一些数值结果验证了理论分析的正确性.     

抛物型方程半离散线性有限元法的不带对数因子|logh|的最大模误差估计

本文考虑抛物型初边值问题的半离散(时间连续)线性有限元格式的最大模误差估计问题.给定     

A multilevel correction method for Stokes eigenvalue problems and its applications

In this paper, a new multilevel correction scheme is proposed to solve Stokes eigenvalue problems by the finite element method. This new scheme contains a series of correction steps, and the accuracy of eigenpair approximation can be improved after each step. In each correction step, we only need to solve a Stokes problem on the corresponding fine finite element space and a Stokes eigenvalue problem on the coarsest finite element space. This correction scheme can improve the efficiency of solving Stokes eigenvalue problems by the finite element method. As applications of this multilevel correction method, a multigrid method and an adaptive finite element technique are introduced for Stokes eigenvalue problems. Some numerical results are given to validate our schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

THE UPWIND FINITE ELEMENT SCHEME AND MAXIMUM PRINCIPLE FOR NONLINEAR CONVECTION-DIFFUSION PROBLEM

In this paper, a kind of partial upwind finite element scheme is studied for twodimensional nonlinear convection-diffusion problem. Nonlinear convection term approximated by partial upwind finite element method considered over a mesh dual to the triangular grid, whereas the nonlinear diffusion term approximated by Galerkin method. A linearized partial upwind finite element scheme and a higher order accuracy scheme are constructed respectively. It is shown that the numerical solutions of these schemes preserve discrete maximum principle. The convergence and error estimate are also given for both schemes under some assumptions. The numerical results show that these partial upwind finite element scheme are feasible and accurate.     

TWO-LEVEL MULTISCALE FINITE ELEMENT METHODS FOR THE STEADY NAVIER-STOKES PROBLEM

In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.     

Two-level multiscale finite element methods for the steady navier-stokes problem

In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.     

Crank–Nicolson Finite Element Scheme and Modified Reduced-Order Scheme for Fractional Sobolev Equation

Abstract

In this paper, a Crank–Nicolson finite difference/finite element method is considered to obtain the numerical solution for a time fractional Sobolev equation. Firstly, the classical finite element method is presented. Stability and error estimation for the fully discrete scheme are rigorously established. However, the amount of calculation and computing time are too large due to many degrees of freedom of classical finite element scheme and nonlocality of fractional differential operator. And then the modified reduced-order finite element scheme with low dimensions and sufficiently high accuracy, which is based on proper orthogonal decomposition technique, is provided. Stability and convergence for the reduced-order scheme are also studied. At last, numerical examples show that the results of numerical computation are consistent with previous theoretical conclusions.     

A stabilized finite element method for the time-dependent Stokes equations based on Crank–Nicolson Scheme

A stabilized finite element method for the time-dependent Stokes equations based on Crank–Nicolson scheme is considered in this paper. The method combines the Crank–Nicolson scheme with a stabilized finite element method which uses the lowest equal-order element pair, i.e., the stabilized finite element method is applied for the spatial approximation and the time discretization is based on the Crank–Nicolson scheme. Moreover, we present optimal error estimates and prove that the scheme is unconditionally stable and convergent. Finally, numerical tests confirm the theoretical results of the presented method.     

A Modified Weak Galerkin Finite Element Method for Sobolev Equation

For Sobolev equation, we present a new numerical scheme based on a modified weak Galerkin finite element method, in which differential operators are approximated by weak forms through the usual integration by parts. In particular, the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element. Optimal order error estimates in discrete $H^1$ and $L^2$ norms are established for the corresponding modified weak Galerkin finite element solutions. Finally, some numerical results are given to verify theoretical results.     

Numerical studies on a novel split-step quadratic B-spline finite element method for the coupled Schrödinger–KdV equations

We propose a novel split-step quadratic B-spline finite element method for solving the initial-boundary value problem of the coupled Schrödinger–KdV equations. A full-discrete finite element scheme is constructed. The conserved properties of the full-discrete scheme are proved. Detailed numerical results show the efficiency of our scheme.     

A monotone scheme for Hamilton–Jacobi equations via the nonstandard finite difference method

A usual way of approximating Hamilton–Jacobi equations is to couple space finite element discretization with time finite difference discretization. This classical approach leads to a severe restriction on the time step size for the scheme to be monotone. In this paper, we couple the finite element method with the nonstandard finite difference method, which is based on Mickens' rule of nonlocal approximation. The scheme obtained in this way is unconditionally monotone. The convergence of the new method is discussed and numerical results that support the theory are provided. Copyright © 2009 John Wiley & Sons, Ltd.     

Solution of double nonlinear problems in porous media by a combined finite volume–finite element algorithm

The combined finite volume–finite element scheme for a double nonlinear parabolic convection-dominated diffusion equation which models the variably saturated flow and contaminant transport problems in porous media is extended. Whereas the convection is approximated by a finite volume method (Multi-Point Flux Approximation), the diffusion is approximated by a finite element method. The scheme is fully implicit and involves a relaxation-regularized algorithm. Due to monotonicity and conservation properties of the approximated scheme and in view of the compactness theorem we show the convergence of the numerical scheme to the weak solution. Our scheme is applied for computing two dimensional examples with different degrees of complexity. The numerical results demonstrate that the proposed scheme gives good performance in convergence and accuracy.