() P. S. Krishnaprasad: Models for Film Growth and Methods for Model Reduction - The process of producing high quality thin (epitaxial, polycrystalline etc.) films of semiconductors (e.g. Si, SiGe) is governed by complex interplay of fluid flow, radiant heating, gas-phase and surface chemistry, and atomic scale phenomena that affect morphology of grown films. Models with predictive power that capture these different elements and provide useful guides for process control are of increasing interest. In this lecture, we discuss the types of models that arise from basic physical considerations of film growth in (commercial) rapid thermal reactors, and explore the problem of passing from these to low dimensional models of practical use in prediction.

The task of producing reduced order models from complex models has a long history in statistical analysis and in control theory under various names such as principal component analysis (PCA), Karhunen-Loeve decomposition, balancing etc.. In the setting of nonlinear systems the techniques are less well-developed, in part due to a lack of effective algorithms for determining the necessary `ignorable' coordinates. In this lecture, we will present some techniques for finding such coordinates. A first illustration of these ideas in connection with pure sensor data will be given (joint work with T. Kugarajah and S. Johnson). This will be followed by a numerical implementation of Morse's Lemma in the critical point theory of functions, for use in model reduction via balancing of nonlinear dynamical systems (joint work with A.J. Newman).

(September 8) Eberhard Bnsch: Numerical methods for the instationary Navier-Stokes equations with a free capillary surface - The instationary Navier--Stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions. A stable finite element discretization is presented. The key idea is the treatment of the curvature terms by a variational formulation. In the context of a discontinuous in time space--time element discretization, stability in (weak) energy norms can be proved.

Numerical examples in 2 and 3 space dimensions are given.

(September 17) Thomas I. Seidman: FEM for a viscoelastic rod model - We discuss the FEM discretization of a viscoelastic model [Seidman/Antman] for longitudinal vibration of a rod. The principal features of interest in the continuum model are (a) avoidance of `infinite compression', (b) consideration of blowup of the elastic response at high compression, and (c) consideration of nonlinear dependence on strain rate. Our concern, then, is to retain these features in the discretization. [This is joint work with D. French and S. Jensen.]

(September 24) Arthur Sherman: Calcium in Tight Spaces - Calcium is both a ubiquitous signal and a toxin in living cells. One way that cells limit the effects of calcium to achieve specific regulation is spatial localization. They exploit the steep gradients near calcium sources, such as ion channels in the cell membrane, supplemented by calcium buffering proteins. We will discuss the computational and analytical challenges of modeling buffered calcium diffusion in nerve terminals and implications for neurotransmitter release.

(October 1) Lawrence Bodin: The Rollon Rolloff Vehicle Routing Problem - In the rollon-rolloff problem, a new sanitation routing problem, tractors move large trailers between locations and a disposal facility. The trailers are so large that the tractor can only transport one trailer at a time. In this talk, the rollon-rolloff problem is defined, a mathematical programming formulation is presented, three heuristic algorithms are developed and tested on over 20 test problems. Conclusions are derived and recommendations for futher research are presented.

(October 8) John Osborn: Can a Finite Element Method Perform Arbitrarily Badly? - We construct elliptic boundary value problems whose standard finite element approximations converge arbitrarily slowly in the energy norm, and show that adaptive procedures cannot improve this slow convergence. We also show that the L₂-norm and the nodal point errors converge arbitrarily slowly. With the L₂-norm two cases need to be distinguished, and the usual duality principle does not characterize the error completely. The constructed elliptic problem is one dimensional.

(October 15) Ricardo Nochetto: Error control of nonlinear evolution equations - We study the backward Euler method with variable time-steps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the angle-bounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual. These estimators solely depend on the discrete solution and data and impose no constraints between consecutive time-steps. We also prove that they converge to zero with an optimal rate w.r.t. the regularity of the solution. We apply the abstract results to a number of concrete strongly nonlinear problems of parabolic type with degenerate or singular character. This is joint work with G. Savare and C. Verdi.

(October 22) Shi Jin: Numerical methods for multiscale transport problems - Many kinetic transport equations have multiple scales that lead to different macroscopic limits. Numerical simulations of there problems become challenging when there are small length scales that are very expensive to resolve. In this talk we will suyvey numerical methods that can overcome the stiffness introduced by the small scales, yet allow the capturing the macroscopic behavior.

(October 29) Haitao Fan: Discrete Shock Profiles for MUSCL Schemes - Shock profiles of numerical schemes for conservation laws epitomize the propagation and structure properties of shocks in numerical solutions. It is also closely related to error estimates of the numerical solutions near shocks. Almost all known results on this subject is for first order monotone schemes and the Lax-Wendroff scheme. The adaptness of higher order schemes makes the analysis difficult. In this talk, we shall establish the existence of discrete shock profiles for a class of monotonicity preserving schemes with continuous flux when s t/x is rational. Examples of schemes in this class are 2nd order flux based MUSCL scheme, the Godunov scheme and all monotonicity preserving schemes when the flux function is C¹ continuous.

(November 5) Daniele Boffi: Mixed finite elements for Maxwell's eigenproblem: the question of spurious modes - In the analysis of the finite element approximation of Maxwell's eigenproblem we are led to the discretization of a linear elliptic problem in mixed form. We shall see that in this case the classical conditions for the well-posedness and stability of mixed approximations are not sufficient to avoid the presence of spurious eigenmodes. We give new sufficient conditions which involve the so-called Fortin operator and apply them to the analysis of the edge element approximation.

(November 12) Mark H. Carpenter: Recent Developements in Finite Difference Methodology - Several topics of current interest are presented. We begin with a discussion on the relative merits of high-order shock capturing schemes. Several different discretization techniques are used to show that all high-order schemes revert to first-order accuracy downstream of a shock, unless special provision is made to account for the shock position. Thus, it is necessary to use both h- and p-refinement in problems having discontinuous solutions. We next present a new approach to deal with complex geometries using multi-domain high-order finite-difference techniques. Special treatments at zonal boundaries guarantee stability, conservation and design accuracy between domains. Several test cases are presented that highlight the new techniques. Finally, we present some new additive (explicit/implicit) Runge-Kutta schemes that show promise in alleviating numerical stiffness in a multi-domain context.

(December 03) Peter S. Bernard: Vortex Method Simulation of Turbulent Flow - A vortex method is described which is specifically designed to be effective in the prediction of turbulent engineering flows. Thus, for example, numerical efficiency is considered to be as important as accurately portraying the flow physics. The algorithm employs smoothed vortex sheets adjacent to solid boundaries to efficiently capture the wall normal vorticity diffusion. Vorticity leaving the sheet region through significant ejection events is converted to vortex tubes, which become the principal means for numerically representing the turbulent field including coherent structures. Chorin's hairpin removal algorithm is used as a natural sub-grid renormalization to prevent runaway production of vortex filaments without harm to the natural cascade of energy to the dissipation range. The effectiveness of the numerical scheme depends on the use of a parallel fast multipole method to evaluate velocities via the Biot-Savart law. Efficient parallelization of this step in a FORTRAN code yields speed up inversely proportional to the number of nodes. Calculations have been made of channel flow and the flow past an impulsively started prolate spheroid. A number of the principal structural and statistical features of these flows are successfully captured. A commercial implementation of the methodology is currently under development.

(December 10) A.Z. Panagiotopoulos: Monte Carlo Simulations of Phase Transitions in Complex Fluids - Significant advances in computational methodologies for phase transitions in fluids have occured in recent years. The Gibbs ensemble Monte Carlo method [AZP, 1987] enables rapid calculations of coexistence among multiple phases in multicomponent systems. More recently, histogram-reweighting methods have enabled highly accurate investigations near critical points. Applications of these methods to formation of micellar aggregates in model surfactant solutions will be presented.