Multigrid methods with nested subspaces and inherited forms are analyzed in an abstract framework that permits application to linear systems of the type that have to be solved at each time level in time-stepping methods for finite element approximations of parabolic problems. Convergence rates that are independent of the space and time steps are obtained in an appropriate time step dependent norm.

In this paper, we construct a three-stage difference scheme of 4th order for ordinary differential equations by the method of composing 2nd order schemes symmetrically.

In this paper the problems to determine an inconsistent system of linear inequalities and to correct its right-hand side vector are solved by using the isometric plane method for linear programming. As an example, the suitable perturbation quantity of the perturbed inequalities of ill-conditioned linear equations is determined in the numerical experiments.

In the author's previous paper [13], a Galerkin/Least-Square type finite element method was proposed and analyzed for the stationary N-S equations. The method is consistent and stable for any combination of discrete velocity and pressure spaces (without requiring the Babuska-Brezzi stability condition). Under the condition that the solution of N-S equations is unique (i.e. in the case of sufficient viscosity or small data), the existence, uniqueness and convergence (at optimal rate) of discrete solution were proved. In this paper, we further investigate the established Galerkin/Least-Square finite element method for the stationary N-S equations. By applying and extending the results of Lopez-Marcos and Sanz-Serna [15], an existence theorem and error estimates are proved in the case of branches of nonsingular solutions.

In this paper, we develop a one-parameter family of P-stable sixth-order and eighth-order two-step methods with minimal phase-lag errors for numerical integration of second order periodic initial value problems: $$ y''=f(t,y), \quad y(t_0)=y_0, \quad y'(t_0)=y'_0. $$ We determine the parameters so that the phase-lag (frequency distortion) of these methods are minimal. The resulting methods are P-stable methods with minimal phase-lag errors. The superiority of our present P-stable methods over the P-stable methods in [1-4] is given by comparative studying of the phase-lag errors and illustrated with numerical examples.

In this paper, first, modified upwind finite element schemes are presented for two-point value problem, and then a class of modified upwind Taylor finite element schemes are derived for one dimensional linear hyperbolic equation. The main point of the paper is how to consider the upwind property to construct base functions to make the schemes obtained be MmB (or TVD). Numerical experiments are given to show that the method is efficient to solve the discontinuous solutions.

In this paper the generalized Newton's method for $LC^1$ unconstrained optimization is investigated. This method is an extension of Newton's method for the smooth optimization. Some basic concepts are introduced according to Clarke(1983). We give optimality conditions for this kind of optimization problems. The local and the global convergence with exact line search are established under the condition of semismoothness.

In this paper, we propose a class of new NCP functions and discuss their properties. By these function, we transfer the complementarity problem into unconstrained optimization problem and study the corresponding optimization problem. Numerical results are given.

We study the sufficient and necessary conditions of the convergence for parameter-based rational methods in a Banach space. We derive a closed form of error bounds in terms of a real parameter $\lambda$ ($1 \leq \lambda < 2$). We also discuss some behaviors when the family is applied to abstract quadratic functions on a Banach space for $ \lambda = 2 $.

In this paper, fully discrete entropy conditions of a class of high resolution schemes with the MmB property are discussed by using the theory of proper discrete entropy flux for the linear scalar conservation laws in two dimensions. The theoretical resluts show that the high resolution schemes satisfying fully discrete entropy conditions with proper discrete entropy flux cannot preserve second order accuracy in the case of two dimensions.