In this paper, we construct a Fourier-Legendre pseudospectral scheme for the unsteady Navier-Stokes equations. This method easily deals with nonlinear terms and saves computational time. The strict error estimations are given.  

In this paper we apply an approach introduced in [6] [7], where continuous norms and high order estimates and extension are used, to study the convergence of vortex methods for the 3-D Euler equations in bounded domains as the initial vorticity $w_0$ and the curl of the body force $f$ are non-compactly supported functions. Convergence results are proved.

Based on a class of functions, which generalize the squared Fischer-Burmeister NCP function and have many desirable properties as the latter function has, we reformulate nonlinear complementarity problem (NCP for short) as an equivalent unconstrained optimization problem, for which we propose a derivative-free descent method in monotone case. We show its global convergence under some mild conditions. If $F$, the function involved in NCP, is $R_0$-function, the optimization problems has bounded level sets. A local property of the merit function is discussed. Finally,we report some numerical results.  

The Ritz vectors obtained by Arnoldi's method may not be good approximations and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the efficiency of Arnoldi type algorithms, we propose a strategy that uses Ritz values obtained from an $m$-dimensional Krylov subspace but chooses modified approximate eigenvectors in an $(m+1)$-dimensional Krylov subspace. Residual norm of each new approximate eigenpair is minimal over the span of the Ritz vector and the $(m+1)$th basis vector, which is available when the $m$-step Arnoldi process is run. The resulting modified $m$-step Arnoldi method is better than the standard $m$-step one in theory and cheaper than the standard $(m+1)$-step one. Based on this strategy, we present a modified $m$-step restarted Arnoldi algorithm.Numerical examples show that the modified $m$-step restarted algorithm and its version with Chebyshev acceleration are often considerably more efficient than the standard $(m+1)$-step restarted ones.  

The abstract $L^2$-norm error estimate of nonconforming finite element method is established. The uniformly $L^2$-norm error estimate is obtained for the nonconforming finite element method for the second order elliptic problem with the lowest regularity, i.e., in the case that the solution $u∈H^1(Ω)$ only. It is also shown that the $L^2$-norm error bound we obtained is one order higher than the energe-norm error bound.  

In this paper we present a family of parallel iterations of order $m+2$ with parameter $m=0,1,...$ for simultaneous finding all zeros of a polynomial without evaluation of derivatives, which includes the well known Weierstrass-Durand-Dochev-Kerner and Börsch-Supan-Nourein iterations as the special cases for $m$=0 and $m$=1, respectively. Some numerical examples are given.  

In this paper, we establish a quasi-Newton method for solving the KKT system arising from variational inequalities. The subproblems of the proposed method are lower-dimensional mixed linear complementarity problems. A suitable line search is introduced. We show that under suitable conditions, the proposed method converges globally and superlinearly.

Let P be a transition matrix of a Markov chain and be of the form $$P=\Bigg( \begin{matrix} P_{11} &P_{12} \\ P_{21} &P_{22}  \end{matrix} \Bigg).$$ The stationary distribution $π^T$ is partitioned conformally in the form $(π^T_1, π^T_2)$. This paper establish the relative error bound in $π^T_i (i=1,2)$ when each block $P_{ij}$ get a small relative perturbation.

In this first paper we present a central relaxing scheme for scalar conservation laws, based on using the local relaxation approximation. Our scheme is obtained without using linear or nonlinear Riemann solvers. A cell entropy inequality is studied for the semidiscrete central relaxing scheme, and a second order MUSCL scheme is shown to be TVD in the zero relaxation limit. The next paper will extend the central relaxing scheme to multi-dimensional systems of conservation laws in curvilinear coordinates, including numerical experiments for 1D and 2D problems.

In the past Index-3 DAEs were solved by BDF methods as multistep methods or implicit Runge-Kutta methods as one-step methods. But if the equations are nonstiff, not only BDF but other multistep methods may be applied. This paper considers four different types of multistep discretization of index 3 DAEs in hessenberg form. The convergence of these methods is proven under the condition that the multistep formula is strictly infinite stable. Numerical tests also confirm the results.