Galerkin Approximations for the Linear Parabolic Equation with the Third Boundary Condition

We solve a linear parabolic equation in ^d , d 1, with the third nonhomogeneous boundary condition using the finite element method for discretization in space, and the -method for discretization in time. The convergence of both, the semidiscrete approximations and the fully discretized ones, is analysed. The proofs are based on a generalization of the idea of the elliptic projection. The rate of convergence is derived also for variable time step-sizes.     

A new approach to discretization applied to a continuous, nonlinear, sliding-mode-like control using nonsmooth analysis

This paper introduces a new approach to discretization of nonlinearcontrol laws with a Lipschitz property. The sampling time isdefined as a parameter, which must be selected sufficientlysmall so that the closed-loop system is stable. In contrastto similar results, the stabilizing effect of the control istaken into account. This can result in less conservative constraintson the minimum sampling frequency. The discretization techniquesare explained on a general nonlinear model and applied to thediscretization of a novel nonlinear, robust sliding-mode-likecontrol law. Similar robustness features as for continuous controlare demonstrated. Nonsmooth Lyapunov functions are used forthe discretized sliding-mode-like control introducing cone shapedregions of the state space. One of these cone shaped regionscoincides with a cone shaped layer around the sliding mode definedby the continuous sliding-mode-like control. A stability theoremusing nonsmooth Lyapunov functions is provided.     

非线性离散微分方程的双曲函数法求解

本文推广了双曲函数方法用于求解非线性离散系统。求解离散的(2+1)维Toda系统和离散的mKdV系统，成功地得到了离散钟型孤立子、离散冲击波型孤立子及一些新的精确行波解。     

A posteriori error estimates of discontinuous Galerkin methods for non-standard Volterra integro-differential equations

^** Email: jingtang{at}lsec.cc.ac.cn^*** Email: hermann{at}math.mun.ca In this paper we establish a posteriori error estimates forthe discontinuous Galerkin (DG) method applied to linear, semilinearand non-standard (non-linear) Volterra integro-differentialequations. We also present an analysis of the DG method withquadrature for the memory term. Numerical experiments basedon three integro-differential equations are used to illustratevarious aspects of the error analysis.     

The Crank–Nicolson finite spectral element method and numerical simulations for 2D non-stationary Navier–Stokes equations

In this paper, we first build a semi-discretized Crank–Nicolson (CN) model about time for the two-dimensional (2D) non-stationary Navier–Stokes equations about vorticity–stream functions and discuss the existence, stability, and convergence of the time semi-discretized CN solutions. And then, we build a fully discretized finite spectral element CN (FSECN) model based on the bilinear trigonometric basic functions on quadrilateral elements for the 2D non-stationary Navier–Stokes equations about the vorticity–stream functions and discuss the existence, stability, and convergence of the FSECN solutions. Finally, we utilize two sets of numerical experiments to check out the correctness of theoretical consequences.     

Modified fully discretized projection method for the incompressible Navier–Stokes equations

The stability and convergence of a second-order fully discretized projection method for the incompressible Navier–Stokes equations is studied. In order to update the pressure field faster, modified fully discretized projection methods are proposed. It results in a nearly second-order method. This method sacrifices a little of accuracy, but it requires much less computations at each time step. It is very appropriate for actual computations. The comparison with other methods for the driven-cavity problem is presented.     

A note on the mesh independence of convergence bounds for additive Schwarz preconditioned GMRES

Additive Schwarz preconditioners, when including a coarse grid correction, are said to be optimal for certain discretized partial differential equations, in the sense that bounds on the convergence of iterative methods are independent of the mesh size h. Cai and Zou (Numer. Linear Algebra Appl. 2002; 9 :379–397) showed with a one‐dimensional example that in the absence of a coarse grid correction the usual GMRES bound has a factor of the order of . In this paper we consider the same example and show that for that example the behavior of the method is not well represented by the above‐mentioned bound: We use an a posteriori bound for GMRES from (SIAM Rev. 2005; 47 :247–272) and show that for that example a relevant factor is bounded by a constant. Furthermore, for a sequence of meshes, the convergence curves for that one‐dimensional example, and for several two‐dimensional model problems, are very close to each other; thus, the number of preconditioned GMRES iterations needed for convergence for a prescribed tolerance remains almost constant. Copyright © 2008 John Wiley & Sons, Ltd.     

New Explicit Rational Solitary Wave Solutions for Discretized mKdV Lattice Equation

In this letter, we study discretized mKdV lattice equation by using a new generalized ansatz. As a result,many explicit rational exact solutions, including some new solitary wave solutions, are obtained by symbolic computation code Maple.     

边界条件对对流扩散方程数值稳定性的影响

本文利用数值计算方法对采用均分网格的一维线性无源的对流－扩散方程在各种边界条件下的稳定性进行了分析,燕求出了不同边界条件下一维问题的中心差分和ＱＵＩＣＫ格式的临界网格Ｐｅｃｌｅｔ数。指出按现有方法得出的临界网格Ｐｅｃｌｅｔ数是判别差分格式对流数值稳定性的最苛刻的要求。对中心差分和ＱＵＩＣＫ格式,除两点边值问题以外的其它边界条件下的稳定性范围均不小于或远远大于两点边值问题的稳定性范围。通过计算还得出了格式的数值稳定性主要取决于计算区域下游侧的边界条件类型而与计算区域上游侧的边界条件类型无关的结论。