n [2--4], symplectic schemes of arbitrary order are constructed by
generating functions. However the construction of generating functions
is dependent on the chosen coordinates. One would like to know that
under what circumstance the construction of generating functions will be
independent of the coordinates. The generating functions are deeply
associated with the conservation laws, so it is important to study their
properties and computations. This paper will begin with the study of
Darboux transformation, then in section 2, a normalization Darboux
transformation will be defined naturally. Every symplectic scheme which
is constructed from Darboux transformation and compatible with the
Hamiltonian equation will satisfy this normalization condition. In
section 3, we will study transformation properties of generator maps and
generating functions. Section 4 will be devoted to the study of the
relationship between the invariance of generating functions and the
generator maps. In section 5, formal symplectic erengy of symplectic
schemes are presented.

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.

In this paper we study a numerical method for the simulation of free surface flows of viscoplastic (Herschel-Bulkley) fluids. The approach is based on the level set method for capturing the free surface evolution and on locally re ned and dynamically adapted octree cartesian staggered grids for the discretization of fluid and level set equations. A regularized model is applied to handle the non-di erentiability of the constitutive relations. We consider an extension of the stable approximation of the Newtonian flow equations on staggered grid to approximate the viscoplastic model and level-set equations if the free boundary evolves and the mesh is dynamically re ned or coarsened. The numerical method is rst validated for a Newtonian case. In this case, the convergence of numerical solutions is observed towards experimental data when the mesh is re ned. Further we compute several 3D viscoplastic Herschel-Bulkley fluid flows over incline planes for the dam-break problem. The qualitative comparison of numerical solutions is done versus experimental investigations. Another numerical example is given by computing the freely oscillating viscoplastic droplet, where the motion of fluid is driven by the surface tension forces. Altogether the considered techniques and algorithms (the level-set method, compact discretizations on dynamically adapted octree cartesian grids, regularization, and the surface tension forces approximation) result in e cient approach to modeling viscoplastic free-surface flows in possibly complex 3D geometries.

A global finite element nonlinear Galerkin method for the penalized Navier-Stokes equations is presented.This method is based on two finite element spaces X_H and X_h,defined respectively on one coarse grid with grid size H and one fine grid with grid size h<0 being the penalty parameter,then two methods are of the same order of approximation.However,the global finite element nonlinear Galerkin method is much cheaper than the standard finite element Galerkin method.In fact,in the finite element Galerkin method the nonlinearity is treated on the fine grid finite element space X_h and while in the global finite element nonlinear Galerkin method the similar nonlinearity is treated on the coarse grid finite element space X_H and only the linearity needs to be treated on the fine grid increment finite element space W_h.Finally,we provide numerical test which shows above results stated.

In this paper, we consider 2D and 3D Darcy-Stokes interface
problems. These equations are related to Brinkman model that treats
both Darcy's law and Stokes equations in a single form of PDE but
with strongly discontinuous viscosity coefficient and zeroth-order
term coefficient. We present three different methods to construct
uniformly stable finite element approximations. The first two
methods are based on the original weak formulations of
Darcy-Stokes-Brinkman equations. In the first method we consider the
existing Stokes elements. We show that a stable Stokes element is
also uniformly stable with respect to the coefficients and the jumps
of Darcy-Stokes-Brinkman equations if and only if the discretely
divergence-free velocity implies almost everywhere divergence-free
one. In the second method we construct uniformly stable elements by
modifying some well-known $H(\Div)$-conforming elements. We give
some new 2D and 3D elements in a unified way. In the last method we
modify the original weak formulation of Darcy-Stokes-Brinkman
equations with a stabilization term. We show that all traditional
stable Stokes elements are uniformly stable with respect to the
coefficients and their jumps under this new formulation.

The convergence problem of the family of Euler-Halley methods is considered under the Lipschitz condition with the L-average, and a united convergence theory with its applications is presented.

Heat transport at the microscale is of vital importace in microtechnology applications. The heat transport equation is different from the traditional heat transport equation since a second order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study,we develop a hybrid finite element-finite difference (FE-FD) scheme weth two levels in time for the three dimensional heat transport equation in a cylindrical thin film with submicroscale thickness. It is shown that the scheme is unconditionally stable. The scheme is then employed to obtain the temperature rise in a sub-microscale cylindrical gold film. The method can be applied to obtain the temperature rise in any thin films with sub-microscale thickness, where the geometry in the planar direction is arbitrary.

This paper deals with analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations with any bounded state-independent lag. Sufficient conditions will be presented to ensurethat any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is dissipative and exponentially stable uniformly for sufficiently small $\epsilon>0$. We will study the numerical solution defined by the linear $\theta-$method and one-leg method and show that they are dissipative and exponentially stable uniformly for sufficiently small $\epsilon>0$ if and only if $\theta=1$.

Some superapproximation and ultra-approximation properties in function, gradient and two-order derivative approximations are shown for the interpolation operator of projection type on two-dimensional domain. Then, we consider the Ritz projection and Ritz-Volterra projection on finite element spaces, and by means of the superapproximation elementary estimates and Green function methods, derive the superconvergence and ultraconvergence error estimates for both prjections, which are also the finite slement approximation solutions of the elliptic problems and the Sobolev equations, respectively.

The bounds for the eigenvalues of the stiffness matrices in the finite element discretization corresponding to Lu := - u" with zero boundary conditions by quadratic hierarchical basis are shown explicitly. The condition number of the resulting system begaves like $O(\frac{1}{h})$ where h is the mesh size. We also analyze a main diagonal preconditioner of the stiffness matrix which reduces the condition number of the preconditioned system to O(1).

The technique described in [4] is used to investigate the analyticity and to obtain second order perturbation expansions of simple non-zero singular values of a matrix analytically dependent on several parameters.

AD (Alternating direction) Galerkin schemes for d-dimensional nonlinear pseudo-hyperbolic equations are studied. By using patch approximation technique, AD procedure is realized, and calculation work is simplified. By using Galerkin approach, highly computational accuracy is kept. By using various priori estimate techniques for differential equations, difficulty coming from non-linearity is treated, and optimal H1 and L2 convergence properties are demonstrated. Moreover, although all the existed AD Galerkin schemes using patch approximation are limited to have only one order accuracy in time increment, yet the schemes formulated in this paper have second order accuracy in it. This implies an essential avancement in AD Galerkin analysis.

Least-squares solution of AXB = D with respect to symmetric positive semidefinite matrix X is considered. By making use of the generalized singular value decomposition, we derive general analytic formulas, and present necessary and sufficient conditions for guaranteeing the existence of the solution. By applying MATLAB 5.2, we give some numerical examples to show the feasibility and accuracy of this construction technique in the finite precision arithmetic.

Superconvergence of the mixed finite element methods for 2-d Maxwell equations is studied in this paper. Two order of superconvergent factor can be obtained for the k-th Nedelec elements on the rectangular meshes.

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.

In this paper, based on the natural boundary reduction advanced by Feng and Yu, we discuss a coupling BEM with FEM for the Dirichlet exterior problems. In this method the finite element grids consist of fine grid and coarse grid so that the singularity at the couner points van be bandled conveniently. In order to solve the coupling system by the preconditioning conjugate gradient method, we construct a simple preconditioner for the "stiffness" matrix. Some error estimates of the corresponding approximate solution and condition number estimate of the preconditioned matrix are also obtained.

We derive new and tight bounds about the eigenvalues and certain sums of the eigen-values for the unique symmetric positive definite solutions of the discrete algebraic Riccati equations. These bounds considerably improve the existing ones and treat the cases that have been not discussed in the literature. Besides, they also result in completions for the available bounds about the extremal eigenvalues and the traces of the solutions of the discrete algebraic Riccati equations. We study the fixed-point iteration methods for com-puting the symmetric positive definite solutions of the discrete algebraic Riccati equations and establish their general convergence theory. By making use of the Schulz iteration to partially avoid computing the matrix inversions, we present effective variants of the fixed-point iterations, prove their monotone convergence and estimate their asymptotic convergence rates. Numerical results show that the modified fixed-point iteration methods are feasible and effective solvers for computing the symmetric positive definite solutions of the discrete algebraic Riccati equations.

We analyze three one parameter families of approximations and show that they are symplectic in Lagrangian sence and can be related to symplectic schemes in Hamiltonian sense by different symplectic mappings. We also give a direct generalization of Veselov variational principle for construction of scheme of higher order differential equations. At last, we present numerical experiments.

In this paper,we discuss a posteriori error estimates of the eigenvalue λ_h given by Adini nonconforming finite element.We give an assymptotically exact error estimator of the λ_h.We prove that the order of convergence of the λ_h is just 2 and the λ_h converge from below for sufficiently small h.

In this paper,we discuss the convergence and superconvergence for eigenvalue problem of the biharmonic equation by using the Hermite bicubic element.Based on asymptotic error expansions and interpolation postprocessing,we gain the following estimation:0≤λ〖TX-〗_h-λ≤C_εh~(8-ε),where ε>0 is an arbitrary small positive number and C_ε>0 is a constant.

The purpose of this paper is to adopt the quasi-interpolating operators in multivariate spline space S~1_2(Δ~(2*)_(mn)) to solve two-dimensional Fredholm Integral Equations of second kind with the hypersingular kernels.The quasi-interpolating operators are put forward in ([7]).Based on the approximation properties of the operators,we obtain the uniform convergence of the approximate solution sequence on the Second Kind Fredholm intergral equation with the Cauchy singular kernel function.

The main purpose of this paper is to derive an explicit expression for Fourier-Chebyshev coefficient $A_{kn}(f)=\frac{\displaystyle 2}{\displaystyle\pi}\int_{-1}^1f(x)T_{kn}(x)\frac{\displaystyle dx}{\displaystyle \sqrt{1-x^2}},k,n\in N_0$,which is initiated by L.Gori andC.A.Micchelli.

In this paper the uniform convergence of Hermite-Fejer interpolation and Griinwald type theorem of higher order on an arbitrary system of nodes are presented.

We present a program for computing symmetric quadrature rules on
triangles and tetrahedra. A set of rules are obtained by using this
program. Quadrature rules up to order 21 on triangles and up to
order 14 on tetrahedra have been obtained which are useful for use
in finite element computations. All rules presented here have
positive weights with points lying within the integration domain.

In this paper we are going to discuss the difference schemes with intrinsic parallelism for the boundary value problem of the two dimensional semilinear parabolic systems. The unconditional stability of the general finite difference schemes with intrinsic parallelism is justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete data of the original problems in the discrete $W^{(1,2)}_2$ norms. Then the uniqueness of the discrete vector solution of this difference scheme follows as the consequence of the stability.

For the system of linear equations arising from discretization of the second-order self-adjoint elliptic Dirichlet-periodic boundary value problems, by making use of the special structure of the coefficient matrix we present a class of combinative preconditioners which are technical combinations of modified incomplete Cholesky factorizations and Sherman- Morrison-Woodbury update. Theoretical analyses show that the condition numbers of the preconditioned matrices can be reduced to $O(h^{-1})$,one order smaller than the condition number O(h^{-2})$ of the original matrix. Numerical implementations show that the resulting preconditioned conjugate gradient methods are feasible, robust and efficient for solving this class of linear systems.

In Part I and Part II of this paper initial-boundary value problems of the acoustic wave equation with absorbing boundary conditions are considered.Their finite element-finite difference computational schemes are proposed.The stability of the schemes is discussed and the corresponding stability conditions are given.Part I and Part II concern the first- and the second-order absorbing boundary conditions,respectively.Finally,numerical results are presented in Part II to show the correctness of theoretical analysis.

A spectral collocation method is proposed to solve Volterra or Fredholm integral equa-tions with weakly singular kernels and corresponding integro-differential equations by virtue of some identities. For a class of functions that satisfy certain regularity condi-tions on a bounded domain, we obtain geometric or supergeometric convergence rate for both types of equations. Numerical results conˉrm our theoretical analysis.

This paper is to develop explicit fourth order symplectic difference schemes for separable Hamiltonian systems.

In this paper, the concept of generalized Nekrasov matrices is introduced, some properties of these matrices are discussed, obtained equivalent representation of generalized diagonally dominant matrices.

In this paper, we investigate multi-scale methods for the inverse modeling in 1-D Metal-Oxide-Silicon (MOS) capacitor. First, the mathematical model of the device is given and the numerical simulation for the forward problem of the model is implemented using finite element method with adaptive moving mesh. Then numerical analysis of these parameters in the model for the inverse problem is presented. Some matrix analysis tools are applied to explore the parameters' sensitivities. And third, the parameters are extracted using Levenberg-Marquardt optimization method. The essential difficulty arises from the effect of multi-scale physical difference of the parameters. We explore the relationship between the parameters' sensitivities and the sequence for optimization, which can seriously affect the final inverse modeling results. An optimal sequence can effivciently overcome the multi-scale problem of these parameters.Numerical experiments show the efficiency of the proposed methods.

In this paper, three versions of WENO schemes WENO-JS, WENO-M and WENO-Z are used for one-dimensional detonation wave simulations with fifth order characteristic based spatial flux reconstruction. Numerical schemes for solving the system of hyperbolic conversation laws using the ZND analytical solution as initial condition are presented. Numerical simulations of one-dimensional detonation wave for both stable and unstable cases are performed. In the stable case with overdrive factor f = 1.8, the temporal histories of peak pressure of the detonation front computed by WENO-JS and WENO-Z reach the theoretical steady state. In comparison, the temporal history of peak pressure computed by the WENO-M scheme fails to reach and oscillates around the theoretical steady state. In the unstable cases with overdrive factors f = 1.6 and f = 1.3, the results of all WENO schemes agree well with each other as the resolution, defined as the number of grid points per half-length of reaction zone, increases. Furthermore, for overdrive factor f = 1.6, the grid convergence study demonstrates that the high order WENO schemes converge faster than other existing lower order schemes such as unsplit scheme, Roe's solver with minmod limiter and Roe's solver with superbee limiter in reaching the predicted peak pressure. For overdrive factor f = 1.3, the temporal history of peak pressure shows an increasingly chaotic behavior even at high resolution. In the case of overdrive factor f = 1.1, in accordance with theoretical studies, an explosion occurs and different WENO schemes leading to this explosion appear at slightly different times.

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.

In this work, a singularly perturbed two-point boundary value problem of convection-diffusion type is considered. An hp version finite element method on a strongly graded piecewise uniform mesh of Shishkin type is used to solve the model problem. With the analytic assumption of the input data,it is shown that the method converges exponentially and the convergence is uniformly valid with respect to the singular perturbation parameter.

Characterizations of symmetric and symplectic Runge-Kutta methods, which are based on the W-transformation of Harier and Wanner, are presented. Using these characterizations we construct two classes of high order symplectic (symmetric and algebraically stable or algebraically stable) Runge-Kutta methods. They include and extend known classes of high order implicit Runge-Kutta methods.

Based on the work of paprer [1], we propose a modified Levenberg-Marquardt algoithm for solving singular system of nonlinear equations $F(x)=0$, where $F(x):R^n\rightarrow R^n$ is continuously differentiable and $F'(x)$ is Lipschitz continuous. The algorithm is equivalent to a trust region algorithm in some sense , and the global convergence result is given. The sequence generated by the algorithm converges to the solution quadratically, if $\|F(x)\|_2$provides a local error bound for the system of nonlinear equations. Numerical results show that the algorithm performs well.

Separable nonlinear least squares problems are a special class of
nonlinear least squares problems, where the objective functions are
linear and nonlinear on different parts of variables. Such problems
have broad applications in practice. Most existing algorithms for
this kind of problems are derived from the variable projection
method proposed by Golub and Pereyra, which utilizes the
separability under a separate framework. However, the methods based
on variable projection strategy would be invalid if there exist some
constraints to the variables, as the real problems always do, even
if the constraint is simply the ball constraint. We present a new
algorithm which is based on a special approximation to the Hessian
by noticing the fact that certain terms of the Hessian can be
derived from the gradient. Our method maintains all the advantages
of variable projection based methods, and moreover it can be
combined with trust region methods easily and can be applied to
general constrained separable nonlinear problems. Convergence
analysis of our method is presented and numerical results are also
reported.