## Session 8: Numerical Analysis Error Estimates and Superconvergence of Mixed Finite Element Methods for Convex Optimal Control Problems

Yanping Chen
Department of Mathematics
Xiangtan University
Xiangtan 411105, Hunan, P.R.China
ypchen@xtu.edu.cn

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.

## A New Trust Region Algorithm for Nonlinear Equations

Jinyan Fan
Department of Mathematics
Shanghai Jiao Tong University
Shanghai 200240, P.R. China
jyfan@sjtu.edu.cn

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.

## Computations of Steady and Unsteady Transport of Pollutant in Shallow Water

Jianhu Feng*
College of Science
Chang'an University
Xi'an, Shaanxi 710064, P.R.China
jhfeng@nwpu.edu.cn

Li Cai
School of Science
Northwestern Polytechnical University
Xi'an, Shaanxi 710072, P.R.China
eign@eyou.com

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.

## Stabilized FE Method for the Stationary N-S Equations

Yinnian He*
Faculty of Science
Xi'an Jiaotong University
Xi'an 710049 P.R. China
heyn@mail.xjtu.edu.cn

Aiwen Wang
School of Basic Courses
Beijing Institute of Machinery
Beijing 100085 P. R. China
wang-aiwen@sohu.com

Liqun Mei
Faculty of Science
Xi'an Jiaotong University
Xi'an 710049 P.R. China

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.

## The Radial Basis Collocation Method for Elliptic Equations with Singularities

Hsin-Yun Hu
National Center for Theoretical Sciences Mathematics Division
National Tsing Hua University
Hsinchu, Taiwan 300
huhy@math.cts.nthu.edu.tw

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.

## On Function Approximation Methods for the Precise Time Integration

Yizhen Huang
School of Electronics Information & Electric Engineering
Shanghai Jiaotong University
Shanghai, 200240, P.R. China
hyz12345678@sjtu.edu.cn

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.

## A Brief Survey of T. Chan's Preconditioner

Xiao-Qing Jin
Department of Mathematics
University of Macau
Macao, P.R. China
xqjin@umac.mo

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 An, T. Chan's circulant preconditioner cF(An) is defined to be the solution of

 min Cn || An - Cn ||F
over all n-by-n circulant matrices Cn. The cF(An), called the optimal circulant preconditioner, has been proved to be a good preconditioner for solving a large class of structured systems.

## Efficient Implementation on Fourier-Chebyshev Collocation Method for Poisson Equation in Non-Cartesian Coordinates

Ming-Chih Lai
Department of Applied Mathematics
National Chiao Tung University
Hsinchu 300, Taiwan
mclai@math.nctu.edu.tw

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.

## Finite Element Methods for Singular Perturbation Problems

Jichun Li
Department of Mathematical Sciences
U.S.A

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.

## A Dimension Split Method for Elastic Shellf

Kaitai Li* and Aixiang Huang
College of Sciences
Xi'an Jiaotong University
Xi'an, 710049, P.R. China
ktli@xjtu.edu.cn

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

 u(x,x) = U(x)+P1Ux+P2Ux2+uk(x,x)x3
where x = (x1,x2) Î w Ì R2,x is a transverse variable.

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)]® E3 is a smooth injective mapping. Let [n\vec] denote a continuously varying unit normal vector along S and let We = w×(-e,e). Hence the set {[^(W)]e} is given by {[^(W)]e} = [(Q)\vec]([(W)]e) where the mapping [(Q)\vec]:[(W)]e Ì R3® E3 is defined by

 ®Q (x1,x2,x) = ®q (x1,x2)+x ®n "(x1,x2,x) Î W e ,   (x1,x2) Î w, ®q (x1,x2) Î S.
Here (x1,x2) is usually called Gaussian Coordinate on S, and (x1,x2,x) is called semigeodesic coordinate system (S-Coordinate System) .S is called middle surface of the shell. The boundary of shell {[^(W)]e } is consists of following:

· top surface Gt = S×{+e},     batten surface Gb = S×{-e},

· lateral surface Gl = 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 aab = [(q)\vec]a[(q)\vec]b,   [(q)\vec]a = [([(q)\vec])/( xa )],   aababl = dal and second and third fundamental forms are given bab = [n\vec][(q)\vec]ab = -[n\vec]a[(q)\vec]b,   cab = [n\vec]a[n\vec]b,   cab = alsbalbbs. Furthermore,as well know contravariant components and inverse matrix of bab,cab are of

 bab = aalabsbls ^b ab bbl = dal,   cab = aalabscls, ^c ab cbl = dal.
(1)
then, we have following relations concerning 1-.2-.3- fundamental forms
 Kaab-2Hbab+cab = 0,   aab-2H ^bab +K ^cab = 0,
(2)
where H,K are mean curvature and Gaussian curvature respectively.
 H = 1 2 aabbab,   K = det(ba) det(aa) = b a ;   caa = aabcab = babbab = 4H2-2K.
(3)

Under coordinate system (xa,a = 1,2,x), the metric tensor of E3 is given gij = [(Q)\vec]i[(Q)\vec]j,   gikgkj = dij. From this it yields that

ì
ï
ï
ï
í
ï
ï
ï
î
 gab(x,x) = aab(x)-2xbab(x)+x2cab(x) = (1-Kx2)aab(x)+2x(Hx-1)bab(x);
 ga3(x,x) = g3a(x,x) = 0,   g33(x,x) = 1,   g(x,x) = det (gij) = q2a(x);
 gab(x,x) = q-2(aab(x)-2K ^bab(x) x+K2x2 ^cab (x));
 g3a(x,x) = ga3(x,x) = 0,   g33(x,x) = 1,    q = 1-2Hx+Kx2
(4)
where x = (x1,x2) . Let Gijk,Ñi, and Gabg,Ña denote Christoffel symbols and covariant derivative in E3 and on S respectively,then they have following relationship[2]
 Glab = * Gl ab +q-1 Rlab;Gab3 = Ga3b = q-1Iab;G3ab = Jab, G333 = G33b = G3b3 = Ga33 = 0
(5)
 Ña ub = * Ñ a ub+q-1(Ibau3+Rbalul), Ñ3u3 = ¶u3 ¶x ; Ñ3ub = ¶ub ¶x +q-1Ibl ul;  Ña u3 = * Ñ a u3+Jalul,
(6)
where
ì
ï
ï
í
ï
ï
î
 Rabl = (2Hx2-x) * Ñ l bab-x2 bam * Ñ l bmb;   Iab = -bab+Kxdab,   Jab = bab-xcab
 Ñi uj = ¶uj ¶xk +Gjikuk, * Ñ a ub = ¶ub ¶xa + * Gb al ul;   ¸u = Ñiuj, * ¸ u = * Ñ a ua
(7)
 eij = 1 2 (Ñiuj+Ñjui) = 1 2 (gjkÑiuk+gikÑjuk)
can be expressed by
ì
ï
ï
í
ï
ï
î
 eab(u)
 = gab(u)-2 1 g ab (u)x+ 2 g ab (u)x2 ;   e33(u) = ¶u3 ¶x ,
 e3a(u)
 = ea3(u) = ga3(u)-bab ¶ub ¶x x+ 1 2 cab ¶ub ¶x x2.
(8)
where
ì
ï
ï
ï
í
ï
ï
ï
î
 gab(u) = 1 2 (aaldsb+abldsa) * Ñ s ul-babu3;   ga3(u) = 1 2 ( * Ñ a u3+aab ¶ub ¶x );
 1 g ab (u) = 1 2 (baldsb+bbldsa) * Ñ s ul- 1 2 cabu3 + 1 2 * Ñ l babul;
 2 g ab (u) = 1 2 (casdlb+cbsdls) * Ñ l us + 1 2 * Ñ l cabul;   rab(u) = 2 1 g ab (u)+ * Ñ a * Ñ b u3.
(9)
Corresponding invariants are
 g0(u) = aabgab(u),   g1(u) = aab 1 g ab (u),   g2(u) = aab 2 g ab (u),   r0(u) = aabrab(u).
 b0(u) = babgab(u), b1(u) = bab 1 g ab (u), b2(u) = bab 2 g ab (u), rb(u) = babrab(u).
(10)
In the sequence we will use following notations:
ì
ï
ï
ï
í
ï
ï
ï
î
 aabst0 = ll0aabast+m(aasabt+aatabs);   babst0 = ll0babbst+m(basbbt+batbbs)
 cabst0 = ll0(aabbbs+aatbab)+m(aasbbt+abtbas+aatbbs+absbst);
 aabst* = ll*aabals+m(aalabs+aasabl);   l* = l l+2m ,   l0 = 2m l+2m ,   l*+l0 = 1
(11)

In this paper we define new linearly change of curvature tensor

 rKTab(u): = rab(u)-l0babg0(u)+2Hgab(u).
(12)
Let us set a 2D-3C(two dimensional and three components) variational problem:
 Find  U(x)  Î  VK(w) such that   £*(U,v0) = < P,v0 >  " v0  Î  VK(w)
(13)
where
 £*(u,v)
 : = óõ óõ w [2aabst0gst(u)gab(v)e+ 2 3 e3(aabst0rKTst(u)rKTab(v)
 -2cabst0(rKTst(u)gab(v)+rKTab(v)gst(u))
(14)
and P is a functional corresponding exterior loading which will be give later.

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

ì
í
î
 -Ñjsij(u)
 = fi        in  W;      ui = 0       on    G0,
 sij(u)nj
 = hi      on    G1;   sij(u)nj = gi      on    GtÇGb.
(15)
where u is a placement vector and f,g,h are give volume force and surface stress tensor acting in the shell and its boundaries respectively. Galerkin variational formulation:
 Find u Î V(W)\doteq{v Î H1(W);v = 0   on   G0}  suchthat    £ (u,v) = < F,v > ,   "v Î V(W)
(16)
where
 £(u,v) = óõ óõ óõ We B(u,v) Öadx1dx2dx,   B(u,v) = qAijklekl(u)eij(v).     (17)
 < F,v >
 = óõ óõ óõ W gijfivjqÖadxdx
 + óõ óõ G1 gijhivjqÖadgdx+ óõ GtÇGb gijgivjq|x = eÈx = -eÖadx
(18)
Aijkl is elasticity tensor defined by (1) and eij(u) is three dimensional strain tensor. We construct an approximate solution to (17) by using solution U of (14) as following
ì
ï
ï
ï
í
ï
ï
ï
î
 UKT(x,x): = U(x)+P1Ux+P2Ux2;
 P1U = -aab * Ñ b U ®e a -l*g0(U) ®n ;
 P2U = -(l*aab * Ñ b g0(U)+2bab * Ñ b U3) ®e a +l*(r0(U)-2Hl0g0(U)-2b0(U)) ®n .
(19)

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.

## A Comparative Study of Block Preconditioners for Incompressible Flow Problems

Michele Benzi and Jia Liu*
Department of Mathematics and Computer Science
Emory University
Atlanta, Georgia 30322, USA
benzi@mathcs.emory.edu and jliu8@mathcs.emory.edu

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.

## The Curvature Relation of Wave Front and Wave Changing in External Field

Shenquan Liu
Department of Mathematics
South China University of Technology
Guangzhou 510640, China
mashqliu@scut.edu.cn

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.

## A Wave Field Decomposition Iterative Method for the Helmholtz Equation in a Waveguide

Ya-Yan Lu
Department of Mathematics
City University of Hong Kong
Kowloon, Hong Kong
mayylu@math.cityu.edu.hk

Abstract : The Helmholtz equation uxx + uzz + 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.

## Chebyshev Spectral Methods on Lattices for High-Dimensional PDEs

Jingtang Ma*
Department of Mathematics
Hong Kong Baptist University
Kowloon Tong, Hong Kong
jingtang@math.hkbu.edu.hk

Dong Li
Program in Applied and Computational Mathematics
Princeton University
Princeton, NJ 08544-1000, U.S.A.
dongli@princeton.edu

Fred J. Hickernell
Department of Mathematics
Hong Kong Baptist University
Kowloon Tong, Hong Kong
fred@math.hkbu.edu.hk

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.

## Predicting Response Surface by Atomic Decomposition

Weichung Wang* and Ray-Bing Chen
Department of Applied Mathematics* and Institute of Statistics
National University of Kaohsiung
Kaohsiung 811, Taiwan
*wwang@nuk.edu.tw and rbchen@nuk.edu.tw

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.

## Some Results on the Condition Numbers

Yimin Wei
Department of Mathematics
Fudan University
Shanghai,200433, P.R. of China
ymwei@fudan.edu.cn

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

## An Algorithm for Computing Steiner Points in Gradient-Constrained Minimum Networksf

D.A. Thomas and J.F. Weng*
ARC Special Research Centre for Ultra-Broadband Information Networks (CUBIN)#
Department of Electrical and Electronic Engineering
The University of Melbourne
Victoria 3010, Australia
d.thomas@ee.mu.oz.au and j.weng@ee.mu.oz.au

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.

## Stopping Criteria in Iterative Methods for Solving Convection-diffusion Equations

Chin-Tien Wu
National Center for Theoretical Sciences
Mathematics Division
National Tsing Hua University, Hsinchu, Taiwan 30043, R.O.C.
ctw@math.cts.nthu.edu.tw

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.

## Approximate Solutions for Parabolic Equations on Rectangular with Reproducing Kernel Function*

Shu-Sen Xie
Department of Mathematics
Ocean University of China
Qingdao, Shandong 266071, P.R.China
shusenxie@ouc.edu.cn

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

## Boundary Moving CHIP Method for Nonconvex Programming

Bo Yu*
Department of Mathematics, Dalian University of Technology
Dalian, Liaoning 116024, P.R. China
yubo@dlut.edu.cn

Yufeng Shang
School of Mathematics, Jilin University
Changchun, Jilin 130012, P.R. China

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.

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## Discrete Green's Function Theory, Locally Symmetric Processing and Superconvergencef

Qi-Ding Zhu* and Jing-Hong Liu
College of Mathematics and Computer Science
Hunan Normal University
Changsha, 410081, Hunan, P.R. China
qdzhu@hunnu.edu.cn

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

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