Abstract : In this talk, we investigate the full discretization of general convex optimal control problems using mixed finite element methods. The state and co-state are discretized by lowest order Raviart-Thomas element and the control is approximated by piecewise constant functions. We derive error estimates for both the control and the state approximation. Moreover, we present the superconvergence analysis for mixed finite element approximation of the optimal control problems.

Abstract : In the traditional trust region algorithm for nonlinear equations, the trust region radius will be larger than a positive constant when the sequence converges to the solution of the problem. In this talk, we present a new trust region algorithm with the trust region converging to zero. The new algorithm has the good property of preventing the trial step from being too large especially for ill-conditioned problems. The convergence rate of the new algorithm is also studied under the local error bound condition which is weaker than the nonsingularity. Finally, some numerical results are given.

Abstract : We present new models for simulating the steady and unsteady transport of pollutant. Then the simple central-upwind schemes based on central weighted essentially non-oscillatory reconstructions are proposed in this paper for computing the one- and two-dimensional steady and unsteady models. Since the nonuniform width of the different local Riemann fans is calculated more accurately, the central-upwind schemes enjoy a much smaller numerical viscosity as well as the staggering between two neighboring sets of grids is avoided. Simultaneously, due to the central-upwind schemes are combined with the fourth-order central weighted essentially non-oscillatory reconstructions, the schemes have the non-oscillatory behavior. The numerical results show the desired accuracy, high-resolution, and robustness of our methods.

Abstract : A stabilized finite element method for the two-dimensional stationary incompressible Navier-Stokes equations is investigated in this work. A macroelement condition is introduced for constructing the local stabilized formulation of the stationary Navier-Stokes equations. By satisfying this condition the stability of the Q₁ - P₀ quadrilateral element and the P₁ - P₀ triangular element are established. Moreover, we obtain the well-posedness and the optimal error estimate of the stabilized finite element method for the stationary Navier-Stokes equations. Finally, we provide some numerical tests to confirm the theoretical results of the stabilized finite element method.

Abstract : This talk presents a meshless and no mapping scheme for seeking the solutions of the pointwise singularity and singular perturbation problems. We introduce the collocation method using radial basis functions as the admissible function. Based on the idea of Ritz-Galerkin method involving approximation quadratures, consequently, we may easily derive the algorithms and theoretical analysis. Some results are given to illustrate the efficiency and accuracy of our method.

Abstract : The precise time integration can give numerical results with extremely high precision for a set of homogeneous ordinary differential equations at integration points. The inaccuracy caused by the approximation to non-homogenous terms is a bottleneck to further promote its precision and efficiency. Originally, Zhong adopted a linear approximation method. In recent years, various efforts have been made to improve this: Lin et al. performed Fourier approximation and Zhou et al. carried out Taylor approximation in his HHPD-T algorithm.

We introduce 4 orthogonal polynomials, the Chebyshev polynomials, the Hermite polynomials, the Laguerre polynomials and the Legendre polynomials to achieve better approximation to the non-homogeneous terms. They are all incorporated with the dimensional expanding technique to avoid computing inverse matrices. It is also proved that it is feasible to incorporate this technique into any polynomial approximation methods. The performance of all 7 methods is tested by both benchmark and complicated numerical examples. Averagely, the least square approximation by Legendre polynomials turns out to be the most excellent for its highest precision under the same expanded dimension and the optimal uniform approximation by Chebyshev polynomials also produces extremely accurate results when the approximating step size is large.

These make precise time integration rather competitive, or even a substitute of their counterparts, such as the h-refinement, p-refinement versions of the traditional Runge-Kutta, Wilson-q or Newmark methods. And the application of error estimation and adaptive techniques to these new precise time integration methods is also a research problem.

Abstract : In this talk, we review some old and develop some new important properties of T. Chan's circulant preconditioner proposed in 1988. For any given n-by-n matrix A_n, T. Chan's circulant preconditioner c_F(A_n) is defined to be the solution of

Abstract : In this talk, we will present an efficient Fourier-Chebyshev collocation method for Poisson-type equation in polar, cylindrical, spherical and elliptical coordinates. In particular, the implementation employs the even-odd parity of Fourier coefficients so that no pole conditions at coordinate singularities are needed. The new implementation is simple and also easy to parallelize.

Abstract : Singular perturbation problems (SPPs) arise in many application areas, such as in chemical kinetics, fluid dynamics and system control, plate and shell problems, etc. Such problems usually contain one or more small parameters in the equations. Solutions of these problems undergo rapid changes within very thin layers near the boundary or inside the problem domain. Such sharp transitions require very fine meshes inside those thin layers to resolve the fine scales.

In this talk, we will first review several numerical techniques developed in the past, especially finite element methods (FEM). Then we introduce some highly non-uniform anisotropic meshe which can be used to solve SPPs efficiently. However, such highly non-uniform mesh complicates the error analysis, which frequently assumes quasi-uniformity in the classical finite element analysis. Here we will present the special techniques, which can be used to prove the global uniform convergence and superconvergence. Finally, numerical experiments supporting the theoretical analysis will be presented.

Abstract : We proposal that the solution u(x,x) to 3D-Elastic shell can be expressed

In this paper, we give the existence for U and provide error estimate of approximation solution.

On the analogy of the asymptotic analysis method proposed by P.G.Ciarlet, we give a similar expansion for the placement u but it can be computed in term by term. Let us consider elastic shells, i.e. of elastic bodies whose reference configuration {[^(W)]_e} Ì E³ (3D-Euclidean space) consists of all points within a distance £ e from a given surface S Ì E³ and e > 0 is thought of as being small. The surface S is defined as the image [(q)\vec] of the closure of a domain w Ì R², where [(q)\vec]: [`(w)]® E<sup>3</sup> is a smooth injective mapping. Let [n\vec] denote a continuously varying unit normal vector along S and let W<sub>e</sub> = w×(-e,e). Hence the set {[^(W)]<sub>e</sub>} is given by {[^(W)]<sub>e</sub>} = [(Q)\vec]([`(W)]_e) where the mapping [(Q)\vec]:[`(W)]_e Ì R³® E³ is defined by

· top surface G_t = S×{+e},     batten surface G_b = S×{-e},

· lateral surface G_l = G₀ÇG₁:G₀ = g₀×{-e,+e}, G₁ = g₁×{-e,+e};,

· boundary of w:g = ¶w,g = g₀Èg₁.

In the following ,Latin indices and exponent:i,j,k ¼,take their values in the set {1,2,3 } while Greek indices and exponents:a,b,g,¼,take their values in the set {1,2}. In addition, Einstain's repeated index summation convention is systematically used.

It is well know, covariant and contravariant component of the metric tensor os the surface of S are given a_ab = [(q)\vec]_a[(q)\vec]_b,   [(q)\vec]_a = [(¶[(q)\vec])/( ¶x^a )],   a^aba_bl = d^a_l and second and third fundamental forms are given b_ab = [n\vec][(q)\vec]_ab = -[n\vec]_a[(q)\vec]_b,   c_ab = [n\vec]_a[n\vec]_b,   c_ab = a^lsb_alb_bs. Furthermore,as well know contravariant components and inverse matrix of b_ab,c_ab are of

Under coordinate system (x^a,a = 1,2,x), the metric tensor of E³ is given g_ij = [(Q)\vec]_i[(Q)\vec]_j,   g^ikg_kj = dⁱ_j. From this it yields that

In this paper we define new linearly change of curvature tensor

It is well known that linear 3-D shell elasticity shows that Boundary value problem^[4]:

Key Words Linear elastic shell, asymptotic expansion method

Subject Classification(AMS): 73L05, 41A60

Reference

[1] Kaitai Li and Aixiang Huang, Mathematical Aspect of the Stream-Function Equations of Compressible Turbomachinery Flows and Their Finite Element Approximation Using optimal Control. Comp. Meth. Appl. Mech. and Eng.41(1983)175-194

[2] Kaitai Li and Aixiang Huang, Tensor Analysis and Its Applications Chinese Scientific Press,2000(in Chinese)

[3] P.G. Ciarlet, Mathematical Elasticity, Vol.III,: Theory of Shells , North-Holland,2000

[4] B.Miara, E. Sanchez-Palencia, Asymptotic Analysis of Linearly Elastic Shells, Asymptotic Analysis 12(1996)41-54

[5] Cristinel Mardare, Asymptotic Analysis of Linearly Elastic Shells: Error Estimates in the Membrane Case, Aymptotic Analysis 17(1998)31-51

[6] Koiter,W.T.,A consistent first approximation in the general theory of thin elastic shells, in Proceedings, IUTAM Symposium on the Theory of Thin Elastic Shells, Delft,August 1959,pp12-33,Amsterdam.

[7] Koiter,W.T.,On the foundations of the linear theory of thin elastic shells,Proc. Kon. Ned.Akad.Wetensch. B73,169-195.

[8] Naghdi, P.M.,Foundations of elastic shell theory,in Progress in Solid Mechanics,Vol.4(I.N.Sneddon and R.Hill,Editors),pp1-90,North-Holland,Amsterdam.

[9] Naghdi,P.M.,The theory of Shells and plates,in Handbuch der Physik,Vol.VIa/2 (S.Fluegge and C.Truesdell, Editors), pp.425-640,Springer-Verlag,Berlin.

[10] Budiansky,B.,Sanders,J.L., On the "best" first-order linear shell theory, in Progress in Applied Mechanics, W.Prager Anniversary Volume(1967),pp129-140,MacMillan, New York.

^fSubsidized by the Special Funds for Major State Basic Research Projects G1999032801 and NSFC10001028.

Abstract : This contribution is concerned with the solution of steady-state incompressible flow problems using preconditioned Krylov subspace methods. Several such preconditioners for the Stokes and Oseen (linearized Navier-Stokes) problems in two and three dimensions are described and experimentally compared. In addition, we consider the Navier-Stokes equations in rotation form. Linearization and application of an implicit time stepping scheme results in a linear problem of Oseen type. Results of several preconditioners for both steady and unsteady Stokes and Oseen problems are presented which illustrate the relative performance of the various preconditioners. In particular we show the excellent performance of the HSS (Hermitian/skew-Hermitian Splitting) preconditioner for the Navier-Stokes equations in rotation form with low viscosity.

Abstract : In this paper, we analyze the influence of external field to the wave structure of excitable media. The theoretical analysis describes the curvature relation of wave front surface in excitable media. The normal velocity of wave front has linear relation with mean curvature, plane velocity and external field. This relation reveals that the normal velocity of wave front will increase in the direction of external field. It gives an explanation to the BZ experiments resulting under temperature field or electric field. The simulation results here show rich spiral wave patterns. In external field, one can see the movement of the whole wave pattern as well. This is consistent with theoretic analysis and BZ experiments phenomena. Simulation results indicate that the spiral wave can move in external field when there is no external stimulus. The moving spiral can disappear immediately or remain breakup for a long period. The breakup results reveal the interaction of multi-spiral, stripe stand wave, water labyrinthian wave and island connection wave pattern [1][2]. For spiral breakup or wave patterns which disappear rapidly, rich patterns results in external stimulus as well. By examining the change of wave, one can find that wave patterns transformation in excitable media is not only the effects of external field but also the results of interaction of many factors. In particular, the boundary shape and the anisotropic nature of the media is also the source for new wave patterns. Due to the complexity of theoretic model, the analysis in this paper did not cover external periodic stimulus, anisotropic media and boundary shape other than rectangle. Rather they are left as future studies.

Abstract : The Helmholtz equation u_xx + u_zz + k₀² n²(x,z) u = 0 is important for many wave propagation problems. Standard numerical methods have difficulties in solving the Helmholtz equation because the resulting linear system is often large, complex, non-Hermitian and indefinite. Without an effective preconditioner, modern Krylov subspace iterative methods may require a large number of iterations. In this talk, re re-formulate the Helmholtz equation as (1 - M) u = u₀, where u₀ is related to the incident waves, M is an implicitly defined integral operator. We apply a Krylov subspace iterative method without a preconditioner to this new equation. An algorithm is developed for fast multiplication of the operator M with any given function. The method will be first developed for a closed waveguide with zero boundary conditions. Efficient implementation of this method depends on the use of discrete sine transform. Numerical results will be presented to demonstrate the performance of this new method.

Abstract : The article studies the Chebyshev spectral methods on using integration lattices for solving the high-dimensional partial differential equations. In this approach, the non-periodic input function is sampled on the node set of an integration lattice. The upper bounds on the error with respect to the weighed L₂ norm and the weighted energy norm are derived. These are independent of the dimensions and therefore avoid the curse of dimensionality, unlike traditional sampling on the grid. Numerical examples are carried out to confirm the theoretical results.

Abstract : We develop algorithms to find so-called effective values x Î Rⁿ, such that the corresponding response f(x) Î R are located in a specific region of interest. Examples of the region of interest include extreme values, bounded intervals, positivity, and others. Following assumptions make the problem challenging: (i) the definition of f(x) is complicated or implicit; (ii) the response surface does not fit to specific patterns; (iii) cost for evaluating the function values is expensive. We iteratively approximate the true yet unknown response surface by simplified surrogate models that are formed by a pre-defined atomics (or bases). The surrogate models are then used to predict the possible effective values. The algorithms are applied to find a positive Lyapunov exponent of the dynamical system. Numerical results show that the algorithms are efficient and practical.

Abstract : In this talk, we will present some recent results on the Moore-Penrose inverse and linear least squares problems.

Abstract : A gradient-constrained minimum network T is a minimum length network spanning a given point set N in Euclidean space with edges whose (absolute) gradients are all no more than an upper bound m. Such networks occur in the mining industry since the tunnels in underground mining networks cannot be very steep: the typical maximum gradient of tunnels is about 1:7 ( » 0.14). To shorten the length of the network T some additional points not in N, called Steiner points, may be added. It has been proved that if m < 1, then the degree of any Steiner point in T is at most 4, and moreover, if m < 0.38, then degree 4 Steiner points are easily determined by solving linear equations. Therefore, the difficulty of constructing a locally minimal gradient-constrained network T is to compute degree 3 Steiner points in T. It also proved that the number of types of degree 3 Steiner points is finite. Suppose s is a locally minimal degree 3 Steiner point in T and its adjacent vertices are a,b,c. In this paper we show that the information from \triangle abc can greatly help us to rule out many infeasible types of s. Moreover, using the variational argument we can further reduce the number of feasible types of s. Based on these considerations we developed an algorithm for computing locally minimal degree 3 Steiner points in gradient-constrained minimum networks. 10,000 (uniformly distributed) random point sets { a,b,c} were tested by this algorithm. The results show that in about 98% of cases s can be determined by solving linear or quadratic equations, and only in about 0.43% of cases s cannot be exactly determined by solving equations and thus certain approximation scheme is essentially required.

^f This research has been supported in part by the Australian Research Council and Newmont Aust. Ltd.

^# CUBIN is an affiliated program of National ICT Australia.

Abstract : The solutions of the convection-diffusion problems often posses sharp gradient layers due to Dirichlet outflow boundaries or discontinuities in boundary conditions. The streamline diffusion finite element methods and adaptive mesh refinement processes are usually employed to overcome such difficulties. In this paper, the a posteriori error estimator based on residual, developed by Verfürth, and the a posteriori estimator based on local solutions, developed by Kay and Silvester, are considered for mesh refinement. The resulting sparse linear systems are solved by iterative methods, such as multigrid methods and Krylov subspace methods. In this talk, we will present some stopping criteria for the iterative solvers such that the iterative errors are bounded by the a posteriori error bounds. Our numerical studies show that the refined meshes obtained from the iterative solutions, which satisfy the proposed stopping criteria, are similar to the refined meshes obtained from the finite element solutions. Furthermore, the multigrid method with the Gauss-Seidel (GS) smoother and the standard linear interpolation requires only half amount of iterations to satisfy our stopping criteria comparing to satisfy the heuristic stopping criterion, the L²-norm of the residual less than 10^-6. However, no such saving can be seen when the generalized minimal residual method (GMRES) with GS preconditioning is used.

Abstract : A numerical method for approximating the solution of parabolic equations using reproducing kernel function is devised and analyzed in this paper. The time discretization are formulated by the Laplace-modification procedure, and the approximate solutions are given as explicit integral expressions using the reproducing kernel function at each time step. The computational advantage of this method is that the schemes are both stable and explicitly solvable, so the computation is full parallel. The stability and error estimates are derived. Some numerical results are presented.

Key words:   parabolic equation, Laplace-modification procedure, reproducing kernel function.

AMS subject classification:   65M60, 65M99

Abstract : In [1] and [2], a homotopy method, called the combined homotopy interior point method (abbr. CHIP method), was presented for solving a class of nonconvex Brouwer fixed point problems and nonconvex programming. Under the so called normal cone condition, probability-one convergence was proven. In [3] and [4], modified versions of the CHIP method for nonconvex programming were given. It was proven that they were probability-one convergent under weaker conditions, however, they are inconvenient for application, because they need to construct some auxiliary map. In [5], another homotopy method for nonconvex programming was given, and probability-one convergence was proven under weaker conditions. The condition is added on the homotopy map and not on the problem to be solved, so it is not clear.

In this paper, motivated by [5], a boundary moving CHIP method is presented. Its probability-one convergence was proven under weaker conditions and, it is more convenient to use than homotopies in [3] and [4].

Reference

[1] B. Yu and Z.H. Lin, Homotopy method for a class of nonconvex Brouwer fixed point problems, Appl. Math. Comput., 74(1996), 65-77.

[2] G.C. Feng, Z.H. Lin and B. Yu, Existence of an interior pathway to a Karush-Kuhn-Tucker point of a nonlinear programming problem, Nonlinear analysis, 32(1998), 761-768.

[3] Q.H. Liu, B. Yu and G.C. Feng, An interior point path-following method for nonconvex programming with quasi normal cone condition, Adv. Math., 29(2000), No.4, 281-282.

[4] B. Yu, Q.H. Liu and G.C. Feng, A combined homotopy interior method for nonconvex programming with pseudo cone condition, Northeast. Math. J., 16(2000), 383-386.

[5] L.T. Watson, Theory for globally convergent probability-one homotopies for nonlinear programming, SIAM J. Optim., 11(2000), 761-780.

Abstract : This paper mainly introduces the Chinese fundamental theories of the research of superconvergence and recent developments, which include the following aspects:

1. the Chinese early achievements of superconvergence

2. the fundamental framework of the Chinese research of superconvergence-'Discrete Green's Function-Two Fundamental Estimates' theory

3. `Discrete Green's Function¡VAsymptotic Expansion' theory

4. `Discrete Green's Function¡VLocally Symmetric Processing' method

5. `Superapproximation¡VLocally Symmetric Processing' and Post-processing Techniques (including SPR and SIR techniques)

6. ultra-approximation and ultraconvergence post-processing techniques

In addition, we will introduce high accuracy algorithms of the multidimensional finite element.

^f Supported by the National Natural Science Foundation of China under Grant 10371038