The Delaunay triangulation, in both classic and more generalized sense, is studied in this paper for minimizing the linear interpolation error (measure in $L^p$-norm) for a given function. The classic Delaunay triangulation can then be characterized as an optimal triangulation that minimizes the interpolation error for the isotropic function $||x||^2$ among all the triangulations with a given set of vertices. For a more general function, a function- dependent Delaunay triangulation is then defined to be an optimal triangulation that minimizes the interpolation error for this function and its construction can be obtained by a simple lifting and projection procedure. The optimal Delaunay triangulation is the one that minimizes the interpolation error among all triangulations with the same number of vertices, i.e. the distribution of vertices are optimized in order to minimize the interpolation error. Such a function- dependent optimal Delaunay triangulation is proved to exist for any given convex continuous function. On an optimal Delaunay triangulation associated with f, it is proved that $\nabla f$ at the interior vertices can be exactly recovered by the function values on its neighboring vertices. Since the optimal Delaunay triangulation is difficult to obtain in practice, the concept of nearly optimal triangulation is introduced and two sufficient conditions are presented for a triangulation to be nearly optimal.

We present a Hermitian and skew-Hermitian splitting (HSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semidefinite matrices. The unconditional convergence of the HSS iteration method is proved and an upper bound on the convergence rate is derived. Moreover, to reduce the computing cost, we establish an inexact variant of the HSS iteration method and analyze its convergence property in detail. Numerical results show that the HSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.

In this paper, we construct the orthogonal wavelet basis in the space of antiperiodic functions by appealing the spline methods. Differing from other results in papers [1,2,3,6,8], here we derive the 3-scale equation, by using this equation we construct some basis functions, those functions can be used to construct different orthonormal basis in some spline function spaces.

In this paper we propose a self-adaptive trust region algorithm. The trust region radius is updated at a variable rate according to the ratio between the actual reduction and the predicted reduction of the objective function, rather than by simply enlarging or reducing the original trust region radius at a constant rate. Weshow that this new algorithm preserves the strong convergence property of traditional trust region methods. Numerical results are also presented.

For a model elliptic boundary value problem we will prove that on strongly regular families of uniform tetrahedral partitions of a pohyhedral domain, the gradient of the quadratic finite element approximation is superclose to the gradient of the quadratic Lagrange interpolant of the exact solution. This supercloseness will be used to construct a post-processing that increases the order of approximation to the gradient in the global $L^2$-norm.

In this paper we show local error estimates for the Galerkin finit element method applied to strongly elliptic pseudo-differential equations on closed curves. In these local estimates the right hand sides are obtained as the sum of a local norm of the residual, which is computable, and additional terms of higher order with respect to the computable, and additional terms of higher order with respect to the meshwidth. Hence, asymptotically, here the residual is an error indicator which provides a corresponding self-adaptive boundary element method.

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.

One-dimensional polynomial interpolation dose not guarantee the convergency and the stability during numerical computation. For two (or multi)-dimensional interpolation, difficulties are much more raising. There are many fundamental problems, which are left open. In this paper,we begin with the discussion of reproducing kernel in two variables. With its help we deduce a two-dimensional interpolation formula. According to this formula, the process of also proven that the error function will decrease monotonically in the norm when the number of knot points is increased. In our formula,knot points may be chosen arbitrarily without any request of regularity about their arrangement. We also do not impose any restriction on the number of knot points. For the case of multi-dimensional interpolation, these features may be important and essential.

This paper suggests a family of parallel iterations with parameter p for finding all roots of a polynomial simultaneously. The convergence of the methods is of order p+2. The methods may also be applied to interval iterations.

In this paper,we will prove the derivative of tetrahedral quadratic finite element approximation is superapproximate to the derivative of the quadratic Lagrange interpolant of the exact solution in the $L^{\infty}$-norm, which can be used to enhance the accuracy of the derivative of tetrahedral quadratic finite element approximation to the derivative of the exact solution.

In this paper, we convert the linear complementarity problem to a system of semismooth nonlinear equations by using smoothing technique. Then we use Levenberg-Marquardt type method to solve this system. Taking advantage of the new results obtained by Dan, Yamashita and Fukushima [11, 33], the global and local superlinear convergence properties of the method are obtained under very mild conditions. Especially, the algorithm is locally superlinearly convergent under the assumption of either strict complementarity or certain nonsingularity. Preliminary numerical experiments are reported to show the efficiency of the algorithm.

Recently, computational results demonstrated remarkable superiority of a so-called "largest-distance" rule and "nested pricing" rule to other major rules commonly used in practice, such as Dantzig's original rule, the steepest-edge rule and Devex rule. Our computational experiments show that the simplex algorithm using a combination of these rules turned out to be even more efficient.

By means of the potential theory Steklov eigenvalue problems are transformed into general eigenvalue problems of boundary integral equations (BIE) with the logarithmic sin gul arity. Using the quadrature rules['], the paper presents quadrature methods for BIE of Steklov eigenvalue problem, which possess high accuracies $O(h^3)$ and low computing complexities. Moreover, an asymptotic expansion of the errors with odd powers is shown. Using $h^3-$Richardson extrapolation, we can not only improve the accuracy order of ap- proximations, but also derive a posterior estimate as adaptive algorithms. The efficiency of the algorithm is illustrated by some examples.

In [4] we proved that all Gauss methods areNT(O)-compatible for neutral delay differential equations (NDDEs) of the form:$$\begin{array}{l} y'(t)=ay(t)+by(t-\tau)+cy'(t-\tau), \ \ t>0, \ y(t)=g(t),\ \ -\tau\leq t\leq 0, \end{array}\tag{0.1}$$ where a, b, c are real, $\tau > 0$, g(t) is a continuous real valued function. In this paper we are going to use the theory of order stars to characterize the asymptotic stability properties of Gauss methods for NDDEs. And then proved that all Gauss methods are $N\tau(0)-$stable.

A new high-order time-stepping finite element method based upon the high-order numerical integration formula is formulated for Sobolev equations, whose computations consist of an iteration procedure coupled with a system of two elliptic equations. Theoptimal and superconvergence error estimates for this new method are derived both in space and in time.Also, a class of new error estimates of convergence and superconvergence for the time-continuous tinite element method is demonstrates can be bounded by the noums of the known data. Moreover, some useful a-posteriori error estimators are given on the basis of the superconvergence estimates.

The thermistor problem is coupled system of nonlinear PDEs which consists of the heat equation with the Joule heating as a source, and the current conservation equation with temperature dependent electrical conductivity. In this paper we make a numerical analysis of the nonsteady thermistor problem.

Iterative Methods are studied for the solution of difference schemes for convection domainated flow problems.

In this paper, a two-scale higher-order finite element
discretization scheme is proposed and analyzed for a Schrodinger
equation on tensor product domains. With the
scheme, the solution of the eigenvalue problem on a fine grid can be
reduced to an eigenvalue problem on a much coarser grid together with some
eigenvalue problems on partially fine grids. It is shown
theoretically and numerically that the proposed two-scale higher-order scheme
not only significantly reduces the number of degrees of freedom but
also produces very accurate approximations.

In this paper, we study adaptive finite element discretization
schemes for an optimal control problem governed by elliptic PDE with
an integral constraint for the state. We derive the equivalent
a posteriori error estimator for the finite element approximation,
which particularly suits adaptive multi-meshes to capture different
singularities of the control and the state. Numerical examples
are presented to demonstrate the efficiency of a posteriori error
estimator and to confirm the theoretical results.

We present a new iterative reconstruction algorithm to improve the
algebraic reconstruction technique (ART) for the Single-Photon Emission
Computed Tomography. Our method is a generalization of the Kaczmarz
iterative algorithm for solving linear systems of equations and
introduces exact and implicit attenuation correction derived from
the attenuated Radon transform operator at each step of the
algorithm. The performances of the presented algorithm have been
tested upon various numerical experiments in presence of both
strongly non-uniform attenuation and incomplete measurements data.
We also tested the ability of our algorithm to handle moderate noisy
data. Simulation studies demonstrate that the proposed method has a
significant improvement in the quality of reconstructed images over
ART. Moreover, convergence speed was improved and stability was
established, facing noisy data, once we incorporate filtration
procedure in our algorithm.

In my earlier paper [4], an eigen-decompositions of the Laplacian operator is given on a unit regular hexagon with periodic boundary conditions. Since an exact decomposition with Dirichlet boundary conditions has not been explored in terms of any elementary form. In this paper, we investigate an approximate eigen-decomposition. The function space, corresponding all eigenfunction, have been decomposed into four orthogonal subspaces. Estimations of the first eight smallest eigenvalues and related orthogonal functions are given. In particulary we obtain an approximate value of the smallest eigenvalue $\lambda_1$～$\frac{29}{40} \pi^2=7.1555$, the absolute error is less than 0.0001.

This paper is concerned with the numerical solution of functional-differential and functional equations which include functional-differential equations of neutral type as special cases. The adaptation of general linear methods is considered. It is proved that A-stable general linear methods can inherit the asymptotic stability of underlying linear systems. Some general results of numerical stability are also given.

We build nite di erence schemes for a class of fully nonlinear parabolic equations. The schemes are polyhedral and grid aligned. While this is a restrictive class of schemes, a wide class of equations are well approximated by equations from this class. For regular (C^2,α ) solutions of uniformly parabolic equations, we also establish of convergence rate of O(α ). A case study along with supporting numerical results is included.

In this paper, we extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by Lions et al. [12] and solve a nonlinear evolutionary parabolic partial differential equation. The fine solver is based on the finite element method or spectral element method in space and a semi-implicit Runge-Kutta scheme in time. The coarse solver is based on a semi-implicit scheme in time and the reduced basis approximation in space. Offline-online procedures are developed, and it is proved that the computational complexity of the on-line stage depends only on the dimension of the reduced basis space (typically small). Parareal in time algorithms based on a multi-grids finite element method and a multi-degrees finite element method are also presented. Some numerical results are reported.

The aim of this paper is to develop a fast numerical method for two-dimensional bound- ary integral equations of the first kind with logarithm kernels when the boundary of the domain is smooth and closed. In this case, the use of the conventional boundary element methods gives linear systems with dense matrix. In this paper, we demonstrate that the dense matrix can be replaced by a sparse one if appropriate graded meshes are used in the quadrature rules. It will be demonstrated that this technique can increase the numerical efficiency significantly.

This paper presents the dual bases for a new family of generalized Ball curves with a position parameter K, which includes the B$\acut{e}zier 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.

This paper is concerned with the investigation of a 2-parametric linear stationary iterative method, called Mofdified Extrapolated Jacobi method, for solving the linear systems Ax=b, where A is nonsingular consistently ordered 2-cyclic matrix.

We study the smooth LU decomposition of a given analytic functional λ-matrix A(λ) and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specifid elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of A(λ), and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.

The well known Wilson’s brick is only convergent for regular cuboid meshes. In this paper a quasi-Wilson element of three dimension is presented which is convergent for any hexahedron meshes. Meanwhile the element is anisotropic, that is it can be used to any flat hexahedron meshes for which the regular condition is unnecessary.

In this paper, we perform a nonlinear multiscale analysis for incompressible Euler equa- tions with rapidly oscillating initial data. The initial condition for velocity field is assumed to have two scales. The fast scale velocity component is periodic and is of order one. One of the important questions is how the two-scale velocity structure propagates in time and whether nonlinear interaction will generate more scales dynamically. By using a La- grangian framework to describe the propagation of small scale solution, we show that the two-scale structure is preserved dynamically. Moreover, we derive a well-posed homoge- nized equation for the incompressible Euler equations. Preliminary numerical experiments are presented to demonstrate that the homogenized equation captures the correct averaged solution of the incompressible Euler equation.

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

Trust region (TR) algorithms are a class of recently developed algorithms for nonlinear optimization. A new family of TR algorithms for unconstrained optimization, which is the extension of the usual TR method, is presented in this paper. When the objective function is bounded below and continuously differentiable, and the norm of the Hese approximations increases at most linearly weth the iteration number, we prove the global convergence of the algorithms. Limited numerical results are reported, which indicate that our new TR algorithm is competitive.

In this paper, a mortar element version for rotated Q1 element is proposed. The optimal error estimate is proven for the rotated Q1 mortar element method.

In this paper, we extend two rectangular elements for Reissner-Mindlin plate [9] to the quadrilateral case. Optimal H and L error bounds independent of the plate hickness are derived under a mild assumption on the mesh partition.

This work concerns the ultraconvergence of quadratic finite element approximations of elliptic boundary value problems. A new, discrete least-squares patch recovery technique is proposed to post-process the solution derivatives. Such recovered derivatives are shown to possess ultraconvergence. The keys in the proof are the asymptotic expansion of the bilinear form for the interpolation error and a “localized” symmetry argument. Numerical results are presented to confirm the analysis.

In this paper, we apply the canonical boundary reduction, suggested by FengKang, to the plane elasticity problems, find the expressions of cannonical integral equations and Poisson integral formulas in some typical domains. We also give the numerical method for solving these equations together with their convergence and error estimates. Coupling with calssical finnite element method, this method can be applied to other domains.

This paper deals with a delay-dependent treatment of linear multistep methods for neutral delay differential equations y'(t) = ay(t) + by(t - τ) + cy'(t - τ), t > 0, y(t) = g(t), -τ≤ t ≤ 0, a,b andc ∈ R. The necessary condition for linear multistep methods to be $N_\tau(0)$-compatible. Figures of stability region for some linear multistep methods are depicted.

The canonical boundary reduction, suggested by Feng Kang, also can be applied to the bidimensional steady Stokes problem. In this paper we first give the representation formula for the solution of the Stokes problem via two complex variable functions. Then by means of complex analysis and the Fourier analysis, we find the expressions of the Poisson integral formulas and the cannoical integral equations in three typical domains. From these results the cannoical boundary elment method for solving the Stokes problem can be developed.