In most domain decomposition (DD) methods, a coarse grid solve is employed to provide the global coupling required to produce an $optimal$ method. The total cost of a method can depend sensitively on the choice of the coarse grid size $H$. In this paper, we give a simple analysis of this phenomenon for a model elliptic problem and a variant of Smith's vertex space domain decomposition method [11, 3]. We derive the optimal value $H_{opt}$, which asymptotically minimizes the total cost of method (number of floating point operations in the sequential case and execution time in the parallel case), for subdomain solvers with different complexities, Using the value of $H_{opt}$, we derive the overall complexity of the DD method, which can be significantly lower than that of the subdomain solver.

This paper is concerned with spectral type of methods using Legendre polynomials. Both Galerkin and collocation approximations for the Navier-Stokes equations are considered and their rates of convergence are obtained. As a consequence, it is shown that these methods achieve spectral accuracy if the solutions to the Navier-Stokes equations are smooth.

A Fourier pseudospectral-finite difference scheme is proposed for unsteady Navier-Stokes equation. It is showed that the numerical solution keeps semi-discrete conservation. The strict error estimation is established. The numerical results are presented.

The nonparametric kernel estimation of probability density function (PDF) provides a uniform and accurate estimate of flood frequency-magnitude relationship. However, the kernel estimate has the disadvantage that the smoothing factor $h$ is estimate empirically and is not locally adjusted, thus possibly resulting in deterioration of density estimate when PDF is not smooth and is heavy-tailed. Such a problem can be alleviated by estimating the density of a transformed random variable, and then taking the inverse transform. A new and efficient circular transform is proposed and investigated in this paper.

Numerical solutions of Riemann problems for 2-D scalar conservation law are given by a second order accurate MmB (local Meximum-minimum Bounds preserving) scheme which is non-splitting. The numerical computations show that the scheme has high resolution and non-oscillatory properties. The results are completely in accordance with the theoretical solutions and all cases are distinguished efficiently.

In this paper, we develop a new technique called multiplicative extrapolation method which is used to construct higher order schemes for ordinary differential equations. We call it a new method because we only see additive extrapolation method before. This new method has a great advantage over additive extrapolation method because it keeps group property. If this method is used to construct higher order schemes from lower symplectic schemes, the higher order ones are also symplectic. First we introduce the concept of adjoint methods and some of their properties. We show that there is a self-adjoint scheme corresponding to every method. With this self-adjoint scheme of lower order, we can construct higher order schemes by multiplicative extrapolation method, which can be used to construct even much higher order schemes. Obviously this constructing process can be continued to get methods of arbitrary even order.

A comparison theorem concerning the regularity of Birkhoff interpolation is given. As an application of this theorem the regularity of $(0,1,…,p-1,p+1,…,M-1,q)$ interpolation $(0<p< M ≤ q)$ is characterized.

In this paper we discuss the convergence of the modified Broyden algorithms. We prove that the algorithms are globally convergent for the continuous differentiable function and the rate of convergence of the algorithms is one-step superlinear and n-step second-order for the uniformly convex objective function. From the discussion of this paper, we may get some convergence properties of the Broyden algorithms.

A new algorithm for unconstrained optimization is developed, by using the product form of the OCSSR1 update. The implementation is especially useful when gradient information is estimated by difference formulae. Preliminary tests show that new algorithm can perform well.