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(January 29) Professor Tien-Mo Shih: Modularized Navier-Stokes Codes - My talk is not intended to present new technical ideas regarding numerical schemes. More than 50% of the content will be somewhat philosophical.

The central theme of my talk is related to distributing among worldwide scientific/engineering (S/E) communities modules that are:

1. easily understood, replaced, improved, inserted, or deleted,
2. of 2nd-order accuracy,

Consequently, S/E users of these modules can readily build up their own personal computer codes (PCC) simply linking these modules, implementing their own parametric values, grid generations, and boundary conditions. The main differences between a commercial computer code (CCC) and a modularized PCC will be discussed. They are: degree of understanding, cost, customization, and user group.

(February 5) Professor Florian Potra: Multibody dynamics with contacts and friction - The field of multi-body dynamics simulation is expected to have a major impact on the design of complex mechanical systems, such as robots and assembly line manipulators. Finding realistic models for impact and friction is very important for accurate simulation. Using the rigid body hypothesis substantially simplifies the model and therefore reduces the complesity of the governing equations. Unfortunately, it has been known for over a century that there are examples of rigid multi-body systems with Coulomb friction which have no mathematical solution in the classical sense. Various authors have proposed different settings in which the rigid multi-body system problem with Coulomb friction has a generalized solution either by allowing for impulsive forces (i.e., a solution in the sense of distributions) or by considering the equations of motion as differential inclusions rather than differential equations. In our talk we present a discrete model that has a computable solution under general conditions and which is very well suited for simulating multi-body systems with friction. A simulation package based on this model is being now implemented at the University of Iowa and some numerical results obtained with a preliminary version of that package

(February 10) Dr. Shari Moskow : A Finite Difference Scheme for Elliptic Equations with Rough Coefficients Using a Cartesian Grid Non-Conforming to Interfaces - A problem of great interest to Schlumberger, an oil services company, is to calculate a potential function in an inhomogeneous medium quickly, easily, and accurately. When the medium varies locally in one direction, we propose a staggered finite difference scheme on a regular Cartesian grid with a special cell averaging. This averaging allows for the change in conductivity to be in any direction with respect to the grid and does not require the grid to be small compared to the layering. We have convergence results and numerical experiments which suggest that the method models thin, nonconformal conductive and resistive sheets well.

(February 12) Dr. Oliver Ernst: On Some Recurrent Theorems Concerning Krylov Subspace Methods - The recent development of Krylov subspace methods for solving linear systems of equations has shown that two fundamental approaches underlie the most commonly used algorithms: the Minimal Residual (MR) and Orthogonal Residual (OR) approaches. We show that these two approaches can be formulated as techniques for solving an approximation problem on a nested sequence of subspaces of a Hilbert space. Many well-known relations among the iterates and residuals of OR/MR pairs are shown to hold even in this abstract setting. When applied to the solution of linear systems of equations and when these subspaces are specialized to be a sequence of Krylov spaces, familiar OR/MR pairs are recovered, among these CR/CG, GMRES/FOM and their specializations MINRES/CG to the Hermitian case, QMR/BCG as well as TFQMR/CGS. Further, we show that a common error analysis for these methods involving the canonical angles between subspaces allows many of the recently developed error bounds to be derived in a simple and uniform manner. An application of this analysis to operator equations involving compact perturbations of the identity shows that OR/MR pairs of Krylov subspace methods converge q-superlinearly for this class of problems.

(February 19) Professor Jian-Guo Liu: Efficient FEM for Unsteady Viscous Incompressible Flow in Vorticity Formulation - We will introduce an efficient time stepping procedure for FEM approximating unsteady incompressible Navier-Stokes equations. For large Reynolds number flows, the convection and viscous terms are both treated explicitly in the vorticity transport equation. The stream function (vector potential in 3D), and hence the velocity, is then evaluated from the vorticity via the kinematic equation. In this new time stepping procedure, the value of the vorticity on the boundary can be obtained explicitly from the steam function (vector potential) without any iteration, thus eliminating some traditional difficulties associated with the vorticity formulation. The resulting schemes are extremely simple and efficient. At each time step, or Runge-Kutta stage, the main computation involves solving a standard Poisson equation and inverting a standard mass matrix. Some theoretical aspects of the method will also been discussed.

(February 26) Professor Peter Monk: Finite Element Methods for Maxwell's Equations - The time harmonic Maxwell equations present a number of challenges to numerical analysts. For example

  • Scattering problems are posed on an infinite domain so that the problem of truncating the solution domain must be faced.
  • The underlying bilinear form is complex, but not Hermitian and non-coercive.
  • The Maxwell system is a vector system with unusual boundary conditions.
This talk will present a biased overview of some of the techniques used to overcome the above problems, and some of the challenges that still must be overcome.

(March 5) Professor Zhiming Chen: Numerical Methods for Dynamical Ginzburg-Landau Vortices in Superconductivity - The phenomenological Ginzburg-Landau complex superconductivitymodel is designed to describe the phenomenon of vortex structure in the superconducting/normal phase transitions. In this talk we report some new development in solving the nonlinear time-dependent Ginzburg-Landau model. The focus will be on a mixed finite element method which calculates the magnetic field accurately, and some results on the a posteriori error estimate which is the basis of developing adaptive finite element methods.

(March 12) Dr. Barbara Wohlmuth: Adaptive Multilevel Techniques for Mixed Finite Element Discretization - During the last couples of years, mixed finite element methods have been increasingly used in applications. In particular, we will consider the two basic ingredients of an efficient adaptive algorithm in detail: Since the arising linear algebraic systems typically become large and ill-conditioned for discretized partial differential equations, there is a need for fast iterative solvers. The construction of the solver is based on an adequate splitting related with the multilevel structure. A further important aspect is the use of adaptively generated triangulations. Local refinement will be performed based on reliable and efficient a posteriori error estimators. A hierarchical error estimator obtained by defect correction in higher order mixed ansatz spaces and a localization by suitable two-level splittings is considered in detail. The performance of the presented algorithm is illustrated by several numerical examples, including a benchmark problem for the neutron fluxes in a light water reactor based on the two-group neutron diffusion equations.

(March 18) Dr. Zhouping Xin: On The Hyperbolic Approximations to First Order Partial Differential Equations and Relaxation Schemes - In this talk, I will discuss some results on a relaxation approximation to some class of nonlinear first order partial differential equations which includes systems of nonlinear hyperbolic conservation laws, Hamilton-Jacobi equations, and mean-curvature flows. The basic idea is to regard the given first order nonlinear systems as the equilibrium for some relaxation process as the rate of relaxation tends to zero. Such an enlarged relaxation system is designed to be linear in convection with a localized stiff source term which are symmetrizable. The numerical advantages of such approximation are amusing. The convergence of such approximation is proved in many interesting cases. The stability condition will be discussed, and some stability results on the approximations in the presence of strong waves such shock waves and boundary layers will be presented. The focus will be the advantages of the numerical methods designed by such a relaxation procedure. Some numerical simulations will be shown.

(March 19) Dr. John C. Strikwerda: Preconditioning for Regular Elliptic Systems of Differential Equations - In solving linear systems of equations arising from discretizations of elliptic systems of equations, it is common to precondition the system so that the resulting system is easier to solve by an iterative method. What this means is that the linear system Ax = b is replaced by the system PA x = Pb , where P, the preconditioning matrix, is chosen to mimic the inverse of A in some way.

If A results for the discretization of an elliptic differential operator E, then one way to choose P is as an approximation to the inverse of a differential operator G so that G^(-1) E is a bounded operator. In this talk I will present a general theory for preconditioning elliptic systems, especially for the Stokes equations and the equations of linear elasticity by using other elliptic systems as preconditioners.

(April 2) Professor G. W. Stewart: The Eigenstructure of Graded Matrices - A matrix A is column graded if it can be written in the form A = BD, where B is a "nice" matrix. For example, the matrix

   6.8563e-01  -1.0540e-04   1.7984e-09   2.3076e-13
  -6.3679e-01  -7.1539e-06  -5.4202e-09   6.7163e-13
  -1.0026e+00   2.7920e-05   1.6342e-08  -5.0808e-13
  -1.8562e-01   1.3733e-04   8.2522e-09   8.5635e-13

is column graded, with a grading ratio of about 1e-3 from column to column.

The eigensystem of a graded matrix generally shares the grading of the matrix. For example, the eigenvalues of the above matrix are

   6.8573e-01  -1.0504e-04   2.3476e-08   2.0918e-12

The eigenvectors

   1.0000e+00   1.5370e-04  -8.1092e-09   5.3325e-13
  -9.2862e-01   1.0000e+00  -3.5687e-05   6.5557e-09
  -1.4621e+00   1.2011e+00   1.0000e+00   5.2613e-05
  -2.7088e-01  -1.0359e+00   2.0688e-01   1.0000e+00

exhibit a more complicated grading, but one that is related to the original.

Unfortunately, this state of affairs is dependent on the "niceness' of the base matrix B. For example, if we change the (2,2)-element of the above matrix to 9.8895e-05, then the eigenvalues become

   6.8573e-01   1.3543e-06  -3.3548e-07   6.9367e-12

and the second eigenvalue is too small.

In this talk we will derive the eigenstructure of graded matrices and give conditions under which base matrix B is nice. Along the way we will review the perturbation theory of eigenspaces--a theory which applies in infinite as well as finite dimensional spaces.

(April 9) Dr. Alison Ramage: Towards Parameter-Free Streamline Upwinding For Advection-Diffusion Problems - The subject of this talk is the design of robust and efficient finite element approximation methods for solving advection-diffusion equations. Specifically, we consider the stabilisation of discrete approximations using uniform grids which do not resolve boundary layers, as might arise using a multi-level (or multigrid) iteration strategy. In particular, we show that when using SUPG (streamline-upwind) finite element methodology, there is a symbiotic relationship between `optimal' solution approximation and fast convergence of smoothers based on the standard GMRES iteration.

(April 16) Professor Charalambos Makridakis: Error estimates for approximations to scalar conservation laws: Relaxation and second-order schemes - A general framework for deriving error estimates for numerical approximations to scalar conservation laws is presented. Applications include finite difference and finite volume relaxation schemes, as well as standard monotone finite volume schemes. Error estimates are derived also for schemes satisfying "strong" as well as "weak" entropy inequalities.

(April 23) Professor Chi-Wang Shu: A Discontinuous Galerkin Finite Element Method for Hamilton-Jacobi Equations - We present a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the Runge-Kutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method.

(April 30) Dr. Z. Jane Wang: Aerodynamics of Insect Flight - Flapping insect flight generates high mean lift by the interaction of the flying surfaces with the shed vorticity. Typical Reynolds numbers of insects are around 5000. The high lift of a hovering insect is not usually explained by conventional quasi-steady aerodynamics. In this study, we compute numerically, for the full Navier-Stokes equations, the unsteady viscous flow that involve vorticity shed from a two-dimensional wing. We model an insect wing by a thin ellipse, which undergoes various translational and rotational motions. We present results describing the vorticity field and the time history of lift and drag forces for various Reynolds numbers and different wing motions. We also discuss computational aspects of the force measurements and the role of the far-field boundary conditions.

(May 7) Professor Celso Grebogi: Obstruction to Shadowing and Modeling - Dynamical conditions for the loss of validity of numerical chaotic solutions of dynamical systems are already understood. However, the fundamental questions of ``how good'' and for ``how long'' the solutions are valid remained unanswered. In this talk, I will address these questions by establishing scaling laws for the shadowing distance and for the shadowing time in terms of meaningful quantities that are easily computable in practice. The scaling theory is verified against a physical model. I will also argue that a new level of mathematical difficulty, brought from the theory of dynamical systems, limits our ability to model nature using deterministic models.