(January 25) Prof. Dr. Eberhard Baensch: Finite Element Discretization for "Nonstandard" Boundary Conditions for the Navier-Stokes Equations - In practical flow computations boundary conditions play a major role to appropriately model certain physical situations. Often these conditions turn out to be of non Dirichlet type such as natural outflow, slip boundary, boundary conditions on time dependent domains or even free boundaries involving surface tension. In the talk several of these conditions and their effective discretization by finite elements are addressed.

(February 1) Prof. R. Bruce Kellogg: Corner singularities, boundary layers, and interior layers - A singularly perturbed convection diffusion equation is considered on a sector. First, the theory of corner singularities is applied to this problem, emphasizing the influence of the singular parameter and the lower order terms. The result is then applied to a sector oriented so as to produce a boundary layer. It is shown how to get an asymptotic expansion of the solution that gives pointwise bounds for the solution and its derivatives in the sector. Finally, we consider a sector oriented so as to produce an interior layer. The results are incomplete here, but the situation is described as far as we know it.

(February 8) Prof. Ricardo H. Nochetto: Positivity preserving finite element interpolation and applications to variational inequalities - Consider the classical problem of interpolating a rough (without point-values) function by continuous piecewise polynomials, but with the additional constraint of preserving positivity. We construct a positive interpolation operator into the space of piecewise linear finite elements with optimal approximation properties. We also give several intriguing impossibility results for such operators concerning boundary values at extreme points, invariance of finite element subspaces, and higher order accuracy. We finally apply positive interpolation to derive optimal a posteriori error estimators for elliptic variational inequalities both in the energy and maximum norms. This work is joint with Z. Chen, K. Siebert, A. Vesser, and L. Wahlbin.

(February 15) Prof. Edward Ott: Two dimensional turbulence with drag - We consider two-dimensional turbulence for the case where the fluid experiences a drag force linear in velocity, as is appropriate to physical situations where two dimensionality applies. We find modifications to the dragless results for the enstrophy cascade k-spectrum power law exponent and predict that (in contrast to the case without drag) intermittency will be present. Our analysis utilizes Lagrangian chaos of the flow. This approach is argued to be valid for the case with drag but not for the case without drag.

(February 22) Omar Lakkis: An a posteriori error estimate and adaptivity for the mean curvature flow of graphs - We are interested in the numerical solution of the equation of motion of a function graph by its mean curvature. We give an a posteriori error analysis of a finite element disretization in spqace of this equation. We then show how to exploit this result for adaptive mesh design.

(March 1) Prof. Georg Dolzmann: Computation of microstructure and relaxation of energies - The analysis of materials undergoing solid to solid phase transformations leads to nonconvex variational problems for the elastic deformation.

Minimizing sequences tend to develop oscillations which model mathematically the microstructures observed in experiments. The appropriate mathematical object for the description of the limit is the so-called gradient Young measure generated by the sequence of deformation gradients.

The numerical approximation of these microstructures is still a challenging, open problem. In this talk, we discuss two aspects in the numerical analysis of nonconvex problems.

In the first part we discuss estimates for finite element minimizers of the original, nonconvex problem. The key idea is to identify the deformation gradient of such a minimizer with a gradient Young measures and to prove estimates in a suitable distance in the space of all nonnegative probability measures.

In the second part we describe advantages and disadvantages of the minimization of the corresponding relaxed functional. In particular, we present a reliable algorithm for the computation of a certain class of gradient Young measures, which can recover the information that is lost by passing from the nonconvex to the relaxed functional.

(March 8) Prof. Chen Wang: Fast Solvers for the 3-D Primitive Equations of the Atmosphere and Ocean Formulated in Mean Pressure Poisson Equation - The three-dimensional primitive equations (PE) of the atmosphere and ocean is taken into consideration. The combination of divergence-free property of the 3-D velocity vector field and the no penetration boundary condition of the vertical velocity on the top and bottom boundary indicates the ``average'' divergence of the horizontal velocity field has to be zero in vertical direction at every $x-y$ plane point. Accordingly, the vertical velocity can be recovered by the integration in $z$ direction of the horizontal divergence. The whole pressure variable is composed of a mean pressure field in horizontal plane, which is determined by a 2-D Poisson equation, along with the integration of the density field in $z$ direction, because of the hydrostatic balance. The Gauge formulation for PE can also be derived, based on the introduction of mean gauge variable in $z$ direction, along with its variation in $z$ direction. The numerical methods based on both formulations are also discussed in detail, including the trick of implementing the physical boundary conditions. 

(March 15) Dr. William G. Szymczak: Simulations of Violent Free Surface Dynamics - When violent free surface motions (such as a breaking wave) occur, regions of "spray" can form. These regions can cause methods which track the free surface to either fail, or at least provide unrealistic results. An algorithm for simulating the long-time behavior of violent free surface motions is presented. This algorithm is based on a generalized foumulation of hydrodynamics which imposes density and pressure constraint onto the conservation equations. Recent extensions to this model include the incorporation of visco plastic effects for modeling saturated sand, and the treatment of virtual mass for predictions of 3D rigid body motions. Comparisons of simulations to both theoretical and experimental results will be presented.

(March 29) Prof. Al Schatz: New asymptotic error expansions in the finite element method and Richardson extrapolation - Some new local asymptotic error expansions for second order elliptic problems in R^N (N \ge 2) will be discussed. This will be used to validate the increase in accuracy of approximations obtained by Richardson extrapolations.

(April 5) Prof. Peter Monk: Analysis of a Finite Element Method for Electromagnetic Scattering - Recently there has been a lot of progress in analyzing finite element methods for Maxwell's equations. I will survey some of this activity and show how to prove convergence for an overlapping boundary element - finite element method for the time harmonic Maxwell system.

(April 12) Prof. Rajarshi Roy: Multiple Wave Interactions in Nonlinear Optical Fibers: Experiments and Models - Optical communications are based on the propagation of waves carrying information through glass fibers. At high intensities, or over long lengths, these waves interact with each other with substantial consequences. Experimental observations and numerical models will be described, including the nonlinear Schroedinger equation and associated ODE systems. Open problems, numerical and physical, will be discussed.

(April 23) Prof. Franco Brezzi: Recent approaches in the treatment of subgrid scales - In a certain number of applications, one has to deal with phenomena that take place on a scale that is smaller than the smallest scale affordable in a finite element discretization. These {\it subgrid} phenomena cannot however be neglected, as their effect on the bigger (computable) scales can be quite relevant. In these cases, the use of parallel computers allows the implementation of suitable strategies, in which a preprocessing is constructed (and executed in parallel, element-by-element) to simulate the effect of the subgrid scales on the computable ones. The lecture shall present a rather general framework in which these pre-processing strategies can be studied.

(April 24) Prof. Franco Brezzi: Discontinuous Galerkin Methods for elliptic problems: a general framework. - At least two different approaches have been used in the literature for introducing Discontinuos Galerkin Methods for diffusion problems. One is based on the use of suitable bilinear forms, possibly with penalty terms at the interelement boundaries (J. Douglas jr, D.N. Arnold, M.F. Wheeler, etc.). The other is based on a finite-volume approach, where a suitable choice is made for the interelement fluxes (Bassi-Rebay, Cockburn-Shu, etc.) The lecture will present a general analysis, in which both approaches can be better studied and understood.

(May 3) Prof. Matthias Gobbert : Mathematical Modeling of Deposition Processes in Semiconductor Manufacturing - The manufacturing of computer chips comprises thousands of individual production steps. One of the fundamental steps consists of achieving a complete fill of small features that have been etched into the surface of the silicon wafer in the previous production step. The most popular process for this purpose is chemical vapor deposition (CVD). However, this process is difficult to control, and uniform deposition becomes hard to attain, in particular as the typical width of features is shrinking below one micrometer. Due to its potential for obtaining uniform coverage even for these features, atomic layer deposition (ALD) has recently gained interest. Both processes will be discussed in this talk. Then I will present a fundamental transport and reaction model for ALD based on the Boltzmann equation. The model is converted to a system of hyperbolic equations by expanding in velocity space, and the discontinuous Galerkin method is used to solve this system. Simulation results for a feature scale ALD model will be presented.

(May 10) Prof. Charles Misner: Bits of numerical relativity -- (1) Spherical Harmonic Decomposition on a cubic grid, (2) Some current progress in numerical relativity at the Albert Einstein Institute, Potsdam, Germany. - (1) A method is described by which a function defined on a cubic grid (as from a finite difference solution of a partial differential equation) can be resolved into spherical harmonic components at some fixed radius. This has applications to the treatment of boundary conditions imposed at radii larger than the size of the grid. (2) Numerical work by the numerical relativity group at the Albert Einstein Institute in Potsdam focuses on predicting the gravitational radiation expected to be emitted by a pair of black holes orbiting each other, which may be the strongest signals reaching the international network of gravitational wave observatories. Recent progress by the group there combines full numerical difference equation approximations to the Einstein equations with linearized approximations to continue waveform predictions beyond the current stability limits of the difference methods.