Approximations using the generalized Laguerre polynomials are investigated in this paper. Error estimates for various orthogonal projections are established. These estimates generalize and improve previously published results on the Laguerre approximations. As an example of applications, a mixed Laguerre-Fourier spectral method for the Helmholtz equation in an exterior domain is analyzed and implemented. The proposed method enjoys optimal error estimates, and with suitable basis functions, leads to a sparse and symmetric linear system.

A numerical method combining the approaches of C.I. Goldstein and L.-A. Ying is used for the simulation in three-dimensional magnetostatics related to an exterior problem in magnetic induction. Recently introduced, this method is based on the use of a graded mesh obtained by gluing homothetic layers in the exterior domain and has been performed in the case of edge element discretizations. In this work, the theoretical and practical aspects of the method are inspected in the case of face element and volume element discretizations, for computing a magnetic induction. Error estimates, implementations, and numerical results are provided.

There exists a strong connection between numerical methods for the integration of ordinary differential equations and optimization problems. In this paper, we try to discover further their links. And we transform unconstrained problems to the equivalent ordinary differential equations and construct the LRKOPT method to solve them by combining the second order singly diagonally implicit Runge-Kutta formulas and line search techniques. Moreover, we analyze the global convergence and the local convergence of the LRKOPT method. Promising numerical results are also reported.

In this paper, finite volume method on unstructured meshes is studied for a parabolic convection-diffusion problem on an open bounded set of $R^d$ ($d = 2$ or $3$) with Robin boundary condition. Upwinding approximations are adapted to treat both the convection term and Robin boundary condition. By directly getting start from the formulation of the finite volume scheme, numerical analysis is done. By using several discrete functional analysis techniques such as summation by parts, discrete norm inequality, et al, the stability and error estimates on the approximate solution are established, existence and uniqueness of the approximate solution and the 1st order temporal norm and $L^2$ and $H^1$ spacial norm convergence properties are obtained.

In this paper, we consider multigrid methods for solving symmetric nonnegative definite matrix equations. We present some interesting features of the multigrid method and prove that the method is convergent in $ L_2 $ space and the convergent solution is unique for such nonnegative definite system and given initial guess.

In this paper we give a convergence theorem for non $C^0$ nonconforming finite element to solve the elliptic fourth order singular perturbation problem. Two such kinds of elements, a nine parameter triangular element and a twelve parameter rectangular element both with double set parameters, are presented. The convergence and numerical results of the two elements are given.  

New estimates are provided for singular values of a matrix in this paper. These results generalize and improve corresponding estimates for singular values in [4]-[6].

It is found that the open composite trapezoidal formulae are the best quadrature formulae under three different senses.

In this paper, we study the relaxed smoothing problems with general closed convex constraints. It is pointed out that such problems can be converted to a convex quadratic minimization problem for which there are good programs in software libraries.

A revised conjugate gradient projection method for nonlinear inequality constrained optimization problems is proposed in the paper, since the search direction is the combination of the conjugate projection gradient and the quasi-Newton direction. It has two merits. The one is that the amount of computation is lower because the gradient matrix only needs to be computed one time at each iteration. The other is that the algorithm is of global convergence and locally superlinear convergence without strict complementary condition under some mild assumptions. In addition, the search direction is explicit.