An alternative to the classical extrapolations is proposed. The stability and the accuracy are studied. The new extrapolation behaves better than the classical ones when there are problems of stability. This technique will be useful in those problems where the region of stability is very small and it forces to work with too fine scales.

In this paper, we discuss the convergence of the Broyden algorithms with revised search direction. Under some inexact line searches, we prove that the algorithms are globally convergent for continuously differentiable functions and the rate of local convergence of the algorithms is one-step superlinear and n-step second-order for uniformly convex objective functions.

In this article we consider a two-level finite element Galerkin method using mixed finite elements for the two-dimensional nonstationary incompressible Navier-Stokes equations. The method yields a $H^1$-optimal velocity approximation and a $L_2$-optimal pressure approximation. The two-level finite element Galerkin method involves solving one small, nonlinear Navier-Stokes problem on the coarse mesh with mesh size $H$, one linear Stokes problem on the fine mesh with mesh size $h << H$. The algorithm we study produces an approximate solution with the optimal, asymptotic in $h$, accuracy.  

 In this article we consider the fully discrete two-level finite element Galerkin method for the two-dimensional nonstationary incompressible Navier-Stokes equations. This method consists in dealing with the fully discrete nonlinear Navier-Stokes problem on a coarse mesh with width $H$ and the fully discrete linear generalized Stokes problem on a fine mesh with width $h << H$. Our results show that if we choose $H=O(h^{1/2}$) this method is as the same stability and convergence as the fully discrete standard finite element Galerkin method which needs dealing with the fully discrete nonlinear Navier-Stokes problem on a fine mesh with width $h$. However, our method is cheaper than the standard fully discrete finite element Galerkin method.  

A practical parallel difference scheme for parabolic equations is constructed as follows: to decompose the domain Ω into some overlapping subdomains, take flux of the last time layer as Neumann boundary conditions for the time layer on inner boundary points of subdomains, solve it with the fully implicit scheme on each subdomain, then take correspondent values of its neighbor subdomains as its values for inner boundary points of each subdomain and mean of its neighbor subdomain and itself at overlapping points. The scheme is unconditionally convergent. Though its truncation error is $O(h+\tau)$the convergent order for the solution can be improved to $O(h^2+\tau)$.

 In this paper, the application of homotopy methods to the load flow multi-solution problems of power systems is introduced. By the generalized Bernshtein theorem, the combinatorial number $C_{2n}^n$ is shown to be the BKK bound of the number of isolated solutions of the polynomial system transformed from load flow equations with generically chosen coefficients. As a result of the general Bezout number, the number of paths being followed is reduced significantly in the practical load flow computation. Finally, the complete P-V cures are obtained by tracking the load flow with homotopy methods.

A filled function is proposed by R.Ge[2] for finding a global minimizer of a function of several continuous variables. In [4], an approach for finding a global integer minimizer of nonlinear function using the above filled function is given. Meanwhile a major obstacle is met: if $ρ > 0$ is small, and $||x_I-\overset{*}{x}_I||$ is large, where $x_I$ - an integer point, $\overset{*}{x}_I$ - a current local integer minimizer, then the value of the filled function almost equals zero. Thus it is difficult to recognize the size of the value of the filled function and can not find the global integer minimizer of nonlinear function. In this paper, two new filled functions are proposed for finding global integer minimizer of nonlinear function, and the new filled function improves some properties of the filled function proposed by R. Ge [2].
Some numerical results are given, which indicate the new filled function (4.1) to find global integer minimizer of nonlinear function is efficient.

This paper presents the dual bases for a new family of generalized Ball curves with a position parameter $K$, which includes the Bézier curve, generalized Said-Ball curve and some intermediate curves. Using the dual bases, the relative Marsden identity, conversion formulas of bases and control points of various curves are obtained.

Klein-Gordon-Schrödinger (KGS) equations are very important in physics. Some papers studied their well-posedness and numerical solution [1-4], and another works investigated the existence of global attractor in $R^n$ and $\Omega ⊂ R^n \ (n\leq 3)$ [5-6,11-12]. In this paper, we discuss the dynamical behavior when we apply spectral method to find numerical approximation for periodic initial value problem of KGS equations. It includes the existence of approximate attractor $A_N$, the upper semi-continuity on $A$ which is a global attractor of initial problem and the upper bounds of Hausdorff and fractal dimensions for $A$ and $A_N$, etc.

A two-grid method for the steady penalized incompressible Navier-Stokes equations is presented. Convergence results are proved. If $h=O(H^{3-s})$ and $\epsilon =O(H^{5-2s}) \ (s=0 \  (n=2); \ s=\frac{1}{2} \ (n=3))$ are chosen, the convergence order of this two-grid method is the same as that of the usual finite element method. Numerical results show that this method is efficient and can save a lot of computation time.

A derivative patch interpolating recovery technique is analyzed for the finite element approximation to the second order elliptic boundary value problems in two dimensional case. It is shown that the convergence rate of the recovered gradient admits superconvergence on the recovered subdomain, and is two order higher than the optimal global convergence rate (ultraconvergence) at an internal node point when even order finite element spaces and local uniform meshes are used.

The purpose of this paper is to study the cascadic multigrid method for the second-order elliptic problems with curved boundary in two-dimension which are discretized by the isoparametric finite element method with numerical integration. We show that the CCG method is accurate with optimal complexity and traditional multigrid smoother (like symmetric Gauss-Seidel, SSOR or damped Jacobi iteration) is accurate with suboptimal complexity.

The purpose of this paper is to construct a kind of multivariate NURBS surfaces by using the bivariate B-splines in the space $S_2^1(\Delta^{(2)}_{mn})$ and discuss some properties of this kind of NURBS surfaces with multiple knots on the type-2 triangulation.