


Mathematics & Statistics

Colloquia — Summer 2002

Friday, June 7, 2002



The Fitting of Response Curve and the Exteme Quantile Estimation in Binary Response Models
Shian Wang
Department of Applied Mathematics
Beijing Institute of Technology
PHY 130
Chris Tsokos and A. N. V. Rao


In the testing of sensitive products, the binary response model is always involved. The most interesting purpose of testing of sensitive products is to fit the response of curve by the data of the same type sensitive products but different lots and to estimate the extreme percentage points.

In 1963, Wetherill suggested the method of transformed response and introduced “Up and Down transformed response” and “One shot transformed response” later.

In 1990, Barry researched the method of “Power Logistic transformed response”. In practical testing of sensitive products, these methods are Generalized.

Wednesday, May 29, 2002



Nonconforming Finite Element Methods
Zhong-Ci Shi
Institute of Computational Mathematics
Chinese Academy of Science
Wen-Xiu Ma


Finite element method has achieved a great success in many fields of sciences and technologies since it was first suggested in elasticity in the fifth decade of the 20th century. Today it becomes a powerful tool for solving partial differential equations.

The key issue of the finite element method is using a discrete solution on the finite element space, usually consisting of piecewise polynomials, to approximate the exact solution on the given space according to a certain kind of variational principle.

When the finite element space is a subspace of the solution space, the method is called CONFORMING. It is known that in this case the finite element solution converges to the true solution provided the finite element space approximates the given space in some sense.

In general, for a \(2m\) order elliptic boundary value problem, the conforming finite element space is a \(C^{m-1}\) subspace. It means that the shape function in this conforming finite element space is continuous together with its \(m-1\) order derivatives. That is, for a second order problem, the shape function is continuous and for a fourth order problem, the shape function and its first derivatives are continuous. It is a rather strong restriction put on the shape functions in the latter case.

Another approach is to relax directly the \(C^{m-1}\) continuity of the finite element space. It comes to the so-called NONCONFORMING finite element method which had and still has a great impact on the development of finite element methods. However, it was found shortly that some nonconforming elements converge and some don't. The convergence behavior sometimes depends on the mesh configuration.

In this talk we will describe and analyze some interesting and important nonconforming finite elements for the 2nd and 4th order elliptic problems. Some of these elements are old and well-known, some are quite new, but all of them are useful in applications.