(September 2) Prof. Dr. Eberhard Baensch: Simulation of dendritic crystal growth with thermal convection - The dendritic growth of crystals under gravity influence shows a strong dependence on convection in the liquid. The situation is modelled by the Stefan problem with Gibbs-Thomson condition coupled with the Navier-Stokes equations in the liquid phase. A finite element method for the numerical simulation of dendritic crystal growth including convection effects is presented. It consists of a parametric finite element method for the evolution of the interface, coupled with finite element solvers for the heat equation and Navier-Stokes equations in a time dependent domain. Results from numerical simulations in two space dimensions with Dirichlet and transparent boundary conditions are included.

(September 3) Prof. Ya-Xiang Yuan: Matrix computation problems in trust region algorithms for optimization - Trust region algorithms are a class of recently developed algorithms for solving optimization problems. The subproblems appeared in trust region algorithms are usually minimizing a quadratic function subject to one or two quadratic constraints. In this talk we review some of the widely used trust region subproblems and some matrix computation problems related to these trust region subproblems.

(September 10) Prof. Dr. Ralf Kornhuber: On monotone multigrid methods for variational inequalities - A wide range of free boundary problems occurring in engineering and industry can be rewritten as a minimization problem for a strictly convex, piecewise smooth but non--differentiable energy functional, or even more, as a variational inequality. The algebraic solution of the related discretized problem is a very delicate question, because usual Newton techniques cannot be applied. We propose a new approach based on convex minimization and constrained Newton type linearization. While convex minimization provides global convergence of the overall iteration, the subsequent constrained Newton type linearization is intended to accelerate the convergence speed. We present a general convergence theory and discuss several applications including the porous medium equation and contact problems in linear elasticity.

(September 16) Prof. Pedro Morin: Data oscillation and convergence of adaptive FEM for elliptic PDE - Data oscillation is intrinsic information missed by the averaging process associated with finite element methods (FEM) regardless of quadrature. Ensuring a reduction rate of data oscillation, together with an error reduction based on a posteriori error estimators, we construct a simple and efficient adaptive FEM for elliptic PDE with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in 2d and 3d yield quasi-optimal meshes along with a competitive performance.

(September 23) Prof. Lars B. Wahlbin: Maximum norm stability in parabolic finite element problems - Consider say the heat equation, u_t=div grad u, in Dx(0,T), with u=0 on bdy D, and u(0)=v. By the maximum principle, max/u(x,t)/ <= max/v(x)/. I shall address the question of similar estimates for finite element approximations, and also the related questions of almost best approximation and convergence in maximum norm.

(September 30) Prof. Zhiming Chen: Finite element methods with matching and non-matching meshes for Maxwell equations with discontinuous coefficients - We investigate the finite element methods for solving time-dependent Maxwell equations with discontinuous coefficients in general three dimensional Lipschitz polyhedral domains. Both matching and non-matching finite element meshes on the interfaces are considered, and optimal error estimates for both cases are obtained. The analysis of the latter case is based on an abstract framework for nested saddle point problems, along with a characterization of the trace space for $H(\curl;D)$, a new extension theorem for $H(\curl;D)$ functions in any Lipschitz domain $D$, and a novel compactness argument for deriving discrete inf-sup conditions. This is a joint work with Qiang Du and Jun Zou.

(October 7) Prof. Tobias von Petersdorff: Local convergence and superconvergence for boundary element methods - We consider an elliptic boundary value problem (e.g., Laplacian or linear elasticity) in a polygon. We discretize it with the boundary element method (BEM) and use graded meshes close to the vertices to compensate vertex singularities in the solution.

We first investigate local convergence rates both away and around the vertices, and how they depend on the strength of the grading of the mesh.

Next, we want to postprocess the BEM solution to obtain the best possible convergence rates in the interior of the domain, close to the boundary, and on the boundary. These optimal convergence rates are higher than the original energy norm convergence rate, and depend on the local convergence results.

(October 14) Mr. Cheng Wang: High order finite difference method for unsteady incompressible flow on multiconnected domain in vorticity-stream function formulation - Using the vorticity and stream-function variables is an effective way to compute 2D incompressible flow due to the fact that the incompressibility constraint for the velocity is automatically satisfied and the pressure variable is eliminated. However, the difficulty arises in the multi-connected computational domain in determining the constants of the stream-function on the boundary of "holes". This is an especially difficult task for the unsteady calculations, since these constants are varying with time to reflect the total fluxes of the flow in each sub channels.

In this talk, we will present an efficient method in finite difference setting to attack this task and present some numerical experiments including clean accuracy, flow past a cooling system, variable density problem, etc. This is joint work with J.-G. Liu.

(October 21) Prof. Manil Suri: Reliability of an hp finite element method for buckling analysis - A linearized model for buckling and stress-stiffening has been recently implemented in the hp code STRESSCHECK. This model does buckling analysis for the fully three-dimensional problem at hand, rather than some asymptotic (dimensionally reduced) limit. It finds the smallest positive multiple q_min of an existing (pre-buckling) stress state s_0 that will result in buckling. The use of the hp method enables solutions over singular domains to be well approximated, and ensures that no locking takes place even when the domain is very thin.

However, a potentially serious danger of the method is that it characterizes q_min as the lowest positive spectral value of a non-compact operator T. Such non-compact spectral value problems can be notoriously ill-behaved, due to the presence of spurious approximate eigenvalues, which can completely pollute the results. We establish that (1) spurious eigenvalues are absent for problems of engineering interest for which the pre-buckling stress s_0 is bounded (2) spurious eigenvalues are always present for problems where s_0 is unbounded. The latter situation occurs because 0 is in the continuous spectrum of the operator T. For this case, we demonstrate how the reliability of the computations can still be assessed, using the eigenvectors (the buckling shapes).

(November 4) Prof. James M. Stone: Numerical methods for astrophysical MHD - Numerical methods are essential tools for studying the dynamics of astrophysical plasmas. In the last decade, a variety of finite-difference methods have been developed for solution of the time-dependent equations of MHD in multi-dimensions. Basic properties of these methods will be discussed, with particular emphasis on techniques for shock-capturing, and for preserving the divergence-free constraint on the numerically evolved magnetic field. Extension of the methods to non-ideal MHD, and to the dynamics of weakly ionized plasmas will be described. The utility of such methods will be demonstrated with several applications performed on parallel supercomputers.

(November 11) Prof. Donald Estep: Estimating the error of numerical solutions of systems of reaction-diffusion equations - One of the pressing problems in the analysis of reaction-diffusion equations is obtaining accurate and reliable estimates of the error of numerical solutions. Recently, we made significant progress using a new approach that at the heart is computational rather than analytical. I will describe a framework for deriving and analyzing a posteriori error estimates, discuss practical details of the implementation of the theory, and illustrate the error estimation using a variety of well-known models. I will also briefly describe an application of the theory to the class of problems that admit invariant rectangles and discuss the preservation of

(November 18) Prof. Eugenia Kalnay: Problems of numerical analysis in weather prediction and data assimilation - I will give an overview of the way operational numerical weather prediction is currently carried out, including global and regional models (with the corresponding problem of boundary conditions), data assimilation (to provide initial conditions for the models), and coupling with ocean and land surface models. I will mention some of the current research areas: (1) development of "national model infrastructures" to allow research and operations to use a common system; (2) variational methods for 4-dimensional data assimilation, with huge matrix inversion and optimization problems; (3) discretization methods for massively parallel computers; (4) development of data assimilation methods for massively parallel computers; (5) assimilation of indirect measurements such as satellite radiances; (6) parameterization of subgrid-scale physical processes such as clouds and radiation.

(December 2) Prof. Zhilin Li: Numerical methods for interface problems and applications - Many physical problems involve interfaces. In this talk, I will introduce our immersed interface method developed for interface problems and recent progress. The methods can be used for the finite difference and the finite element methods based on Cartesian grids. Our methods can handle both discontinuous coefficients and singular sources. The main idea is to incorporate the known jumps in the solution and its derivatives into the numerical schemes. Some numerical results and applications will also be presented in the talk.

(December 9) Prof. Douglas Arnold: Approximation by quadrilateral finite elements - Finite element spaces are usually defined by a three step process. First, a finite dimensional space of functions, often polynomial, is given on a reference element, such as the unit simplex or cube. Second, this space is transferred to arbitrary images of the reference element via transformations of a given class, e.g., affine or multilinear. Finally, the finite element space is pieced together on a mesh consisting of transformed images of the reference elements. A fundamental issue is to determine the rate of approximation afforded by the resulting finite element space as the mesh is refined. Assuming that the space given on the reference element contains all polynomials of a certain degree and the transformations from the reference element are affine, the theory establishing optimal rates of approximation is one of the pillars of the mathematical foundations of finite elements. However the situation is more complicated for non-affine transformations, such as are needed for quadrilateral meshes, and the results in this case are--suprisingly--not widely known. In this talk I will review the classical theory of finite element approximation and prove a new result establishing necessary and sufficient conditions for optimal order quadrilateral approximation. One consequence is that many popular finite element schemes attain lower rates of convergence on quadrilateral meshes than has been generally believed.