(January 31) Professor Victor Nistor: Boundary Value Problems with Distributional Data and their Numerical Approximation - I will begin by providing four formulations for the Laplace equation on a smooth, bounded domain with the boundary data given by a distribution. The weak formulation is immediately seen to satisfy the `inf-sup' condition of Babuska, Brezzi, and Ladyzhenskaya. This proves the well posedness of these problems and provides a priori estimates for the solution. These a priori estimates extend to the other three formulations because all four formulations of the Laplace equation with distributional data are equivalent.

The weak formulation for the Neumann problem allows us to define discrete solutions in Generalized Finite Element Space. Interior estimates of discrete harmonic functions (due to Nitsche-Schatz) allow us also to obtain interior error estimates for our discrete solutions. These results are joint work with Ivo Babuska. In the end, I will also describe some recent results with Ivo Babuska and Nicolae Tarfulea on the interior approximation of solutions of the Dirichlet problem for the Laplace equation with distributional boundary conditions.

(February 3) Professor Felix Otto: Multiscale Analysis in Micromagnetics - From the point of view of mathematics, micromagnetics is an ideal playground for a pattern forming system in materials science: There are abundant experiments on a wealth of visually attractive phenomena and there is a well--accepted continuum model. In this talk, I will focus on two specific experimental pattern for thin film ferromagnetic elements. One pattern is a ground state, the other pattern is a metastable state. Starting point for our analysis is the micromagnetic model which has three length scales and thus many parameter regimes. For both pattern, we identify the appropriate paramater regime and rigorously derive a reduced model via $\Gamma$--convergence. We numerically simulate the reduced model and compare to experimental data. This is joint work with A. DeSimone, R. V. Kohn, and S. Mueuller

(February 14) Prof. Michele Benzi: Robust Iterative Solvers for the Navier-Stokes Equations - Constraint enforcement is a method to preserve invariants such as energy and momentum in the time evolution of a physical process. The usual approach using Lagrange multipliers leads to a matrix with a characteristic block form, often called a ``saddle point matrix.'' Linear systems with a saddle point matrix are hard to solve.

We will present recent work on solving these systems when they arise from the Navier-Stokes equations, in both the steady and unsteady flow cases. We compare Picard-type linearizations for the standard form of the convection term with alternative linearizations using the rotation form.

Our specific interest is in preconditioning methods. We focus on block triangular preconditioners based on an Augmented Lagrangian formulation of the discrete saddle point problem and on preconditioners derived from the Hermitian/Skew-Hermitian splitting.

We examine the performance of various solvers as the mesh size, Reynolds number, time step, and other problem parameters vary. Our results show that fast convergence is achieved in many cases, independent of problem parameters.

This is joint work with Maxim Olshanskii (Moscow State University) and with my PhD student, Jia Liu.

(February 21) Professor Al Schatz: Maximum Norm Estimates in the Finite Element Method Allowing Highly Refined Grids - I will describe some joint work with R. Nochetto and J. Xu. Roughly speaking, we shall give conditions on the mesh so that the finite element method, for second order elliptic problems on plane polygonal domains, have an "almost" optimal rate of convergence for as large a class of meshes as possible.

(February 28) Professor Beatrice Riviere: On the Solution of Complex Flow and Transport Processes - n this talk, we present high order numerical methods for solving multiphysics problems. First, we investigate the coupling of surface flow with subsurface flow. Surface flow is characterized either by Stokes or Navier-Stokes equations whereas subsurface flow is characterized by Darcy equations. Special interface conditions are considered between the subregions. Locally conservative methods, such as discontinuous Galerkin or mixed finite element methods, are used. Optimal error estimates are obtained. Second, we consider both implicit and explicit discontinuous formulations of the incompressible two-phase flow problem. In particular, we show numerical convergence of the p-version. An interesting fact is that no slope limiters are needed for the implicit method.

(March 7) Profesor Yousef Saad: Solving Large Scale Eigenvalue Problems in Electronic Structure Calculations - Density Functional Theory (DFT) is a successful technique used to determine the electronic structure of matter which is based on a number of approximations. It converts the original $n$-particle problem into an effective one-electron system, resulting in a coupled one-electron Schr\"odinger equation and a Poisson's equation. This coupling is nonlinear and rather complex. It involves a charge density $\rho$ which can be computed from the wavefunctions $\psi_i$, for all occupied states. However, the wavefunctions $\psi_i$ are the solution of the eigenvalue problem resulting from Schr\"odinger's equation whose coefficients depend nonlinearly on the charge density. This gives rise to a non-linear eigenvalue problem which is solved iteratively. The challenge comes from the large number of eigenfunctions to be computed for realistic systems with, say, hundreds or thousands of electrons. We will discuss a parallel implementation a finite difference approach for this problem with an emphasis on diagonalization. We will illustrate the techniques with our in-house code, called PARSEC. This code has evolved over more than a decade as features were progressively added and the diagonalization routine, which accounts for the biggest part of a typical execution time, was upgraded several times. We found that it is important to consider the problem as one of computing an invariant subspace in the non-linear context of the Kohn-Sham equations. This viewpoint leads to considerable savings as it de-emphasizes the accurate computation of individual eigenvectors and focuses instead on the subspace which they span.

(March 14) Professor Uday Banerjee: Superconvergence & Recovery in the Generalized Finite Element Method - Superconvergence is an important and well known feature of the classical finite element method. This feature allows accurate approximation of the derivatives of the solution of the underlying boundary value problem, which can be used in post-processing. In this talk, we will briefly describe the Generalized Finite Element Method and show that it also has the superconvergence property. We will also present some computational results, which will indicate that the ``recovered derivatives'' -- constructed from the computed solution -- yield very good approximation of the derivatives of the exact solution.

(March 28) Professor James Nagy: Enforcing Nonnegativity for Ill-Posed Problems in Image Processing - Ill-posed problems arise in many image processing applications, including microscopy, medicine and astronomy. Iterative methods are typically recommended for these large scale problems, but they can be difficult to use in practice. For example, enforcing nonnegativity constraints can be expensive. In this talk we consider a class of efficient nonnegatively constrained iterative algorithms that are derived from the noise statistics of the image formation process, and show that these can be interpreted as covariance preconditioned steepest descent methods. Image processing examples are used to illustrate concepts and to test and compare algorithms.

(April 4) Professor Susanne Brenner: Fast Solvers for C⁰ Interior Penalty Methods - C⁰ interior penalty methods are discontinuous Galerkin methods for fourth order elliptic boundary value problems that have many advantages. In this talk we will first give a brief introduction to C⁰ interior penalty methods and then discuss multigrid and domain decomposition methods for solving the resulting systems. We will present convergence results for the V-cycle, W-cycle and F-cycle multigrid algorithms, and also condition number estimates for two-level additive Schwarz preconditioners. Numerical results will also be reported. also be reported.

(April 11) Professor Randolph Bank: A Domain Decomposition Solver for a Parallel Adaptive Meshing Paradigm - We describe a domain decomposition algorithm for use in the parallel adaptive meshing paradigm of Bank and Holst. Our algorithm has low communication, makes extensive use of existing sequential solvers, and exploits in several important ways data generated as part of the adaptive meshing paradigm. Numerical examples illustrate the effectiveness of the rocedure.

(April 20) Profesor Yann Brenier: Derivation of Particle, String and Membrane Motions from Born-Infeld Electromagnetism - We derive classical particle, string and membrane motion equations from a rigorous asymptotic analysis of the Born-Infeld nonlinear electromagnetic theory. The Born-Infeld model is a modification of the classical Maxwell equations, designed in 1934 to cutoff, in a nonlinear fashion, the infinite electrostatic field generated by point charged particles, at a scale of order 10 to the -15 meters. Quickly shadowed by the success of Quantum Electrodynamics in the 40', this model has known a recent revival through the concept of Dirichlet-Branes in String Theory. Our asymptotic analysis starts by adding to the Born-Infeld equations the corresponding energy-momentum conservation laws and write the resulting system as a non-conservative symmetric system of ten first-order evolution PDEs. (Surprisingly enough, the extended system enjoys Galilean invariance, just like classical Fluid Mechanics.) Then, we show that four rescaled versions of the system have smooth solutions existing in the time interval where the corresponding limit problems have smooth solutions. These limits respectively describe the classical linear Maxwell equations (for weak fields), string motions (for large magnetic fields and moderate electric fields), particle motions (for large electromagnetic fields) and membrane motions. This is partly a joint work with Wen-An Yong, from Heidelberg.

(April 21) Professor Yann Brenier: String Integration of Some MHD Equations - We first review the link between strings and some Magnetohydrodynamics equations. Typical examples are the Born-Infeld system, the Chaplygin gas equations and the shallow water MHD model. They arise in Physics at very different (from subatomic to cosmologic) scales. These models can be exactly integrated in one space dimension by solving the 1D wave equation and using the d'Alembert formula. We show how an elementary "string integrator" can be used to solve these MHD equations through dimensional splitting. A good control of the energy conservation is needed due to the repeted use of Lagrangian to Eulerian grid projections. Numerical simulations A good control of the energy conservation is needed due to the repeted use of Lagrangian to Eulerian grid projections. Numerical simulations in 1 and 2 dimensions will be shown.

(April 25) Professor Jinchao Xu: Robust Iterative Methods for Nearly Singular, Indefinite and H(curl)/H(div) Systems - Two related results will be presented in this talk. First, a new theory will be presented on the method of subspace corrections for nearly singular systems. Seconly, a new class of optimal solvers will be given and analyzed for H(curl) and H(div) systems in terms of Poisson Solvers. As a consequence, optimal AMG solvers can be obtained for H(curl) and H(div) systems whenever optimal AMG solvers are available for Poisson equations. The new methods can be naturally applied to Maxwell equations, Stokes equations and mixed finite element methods. Both theoretical and numerical examples will be presented.

(May 2) Dr. Rich Lehoucq: Discrete Dynamical Systems and Iterative Eigensolvers - We present an isomorphism between discrete dynamical systems and a class of iterations for the eigenvalue problem. This isomorphism provides a framework to investigate the convergence and stability of existing eigen-iterations and propose new techniques. The isomorphism with discrete dynamical systems allows us to consider geometric properties. In particular, we demonstrate that there are first integrals, or invariants, that must be satisfied by the eigen-iteration. Satisfaction of these invariants is demonstrated to be crucial to the numerical stability of the eigen-iteration. Moreover, these invariants provide a mechanism for monitoring the stability of the eigen-iteration. This is joint work with Mark Embree, Rice University.

(May 9) Profesor Bernardo Cockburn: An Adaptive Method for Hamilton-Jacobi Equations - In this talk we review our work on adaptive algorithms for Hamilton-Jacobi equations. We consider an adaptive version of the discontinuous Galerkin method for Hamilton-Jacobi equations proposed by Hu and Shu in 1999. It works as follows. Given the tolerance and the degree of the polynomial approximation of the approximate solution, the adaptive algorithm finds a mesh on which the approximate solution has a distance (in the unifrom norm) to the viscosity solution no bigger than the prescribed tolerance. The algorithm uses two main tools. The first is an a posteriori error estimate which is the main contribution of our work. The second is a new method that allows us to find a new mesh as a function of the old mesh and the ratio of the a posteriori error estimate to the tolerance. We display extensive numerical evidence that indicates that, for any given polynomial degree, the method achieves its goal with optimal complexity independently of the tolerance.

This is joint work with Bayram Yenikaya (Sillicon Valley) and Yanlai Chen (U. of Minnesota).