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(January 21) Ian Sloan: Good approximations on the sphere, with applications to geodesy and the scattering of sound - The theme of this talk is that polynomial approximations on the sphere are important for applications, but that successful applications of high-degree polynomials need a good understanding of underlying approximation properties. We illustrate with two case studies.

First, for applications in geodesy, there is good reason to use cubature rules that have a high degree of polynomial accuracy. The stability, and even the computability, of such rules depends critically on the properties of the underlying polynomial interpolants. Second, a recent spectral approach to the scattering of sound by three dimensional objects needs for its analysis good approximation properties of the hyperinterpolation polynomial approximation scheme. In the course of this talk the existing state of knowledge for both interpolation and hyperinterpolation will be reviewed.

(January 28) Georgios Zouraris: Finite element methods for elliptic PDEs with stochastic coefficients - We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. The aim of the computations is to approximate the expected value of the solution. The first method generates iid approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The second method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, or either $h$ or $p$ version, then approximates the corresponding deterministic solution yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the asymptotic computational work required by each numerical method to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical method.

(February 4) Pedro Morin: Finite Element Methods for Surface Diffusion - We consider the evolution of a 1- or 2-dimensional surface by surface diffusion. That is, the normal velocity of the surface is given by the Surface Laplacian of the Mean Curvature.

We introduce a weak formulation of the 4th order equation, based on a mixed approach, which converts the system into four lower order PDE. We propose a finite element discretization, show stability and conservation properties, and present several numerical experiments.

The discretization and discussion will be done for two different cases: parametric surfaces and surfaces which are the graph of a function.

(February 11) Andreas Veeser: A posteriori error estimates, averaging, and superconvergence - Many finite element packages incorporate a post-processing procedure for smoothing the discontinuous discrete gradient. For this reason, an algorithmically convenient approach to a posteriori error estimation is based on measuring the difference between the direct and post-processed (typically averaged) gradient.

For a simple model problem, we will give a modification of such an error estimator, which requires some additional computational work. However, the latter pays off, since the modified estimator

  • is not only asymptotically but unconditionally reliable and efficient,
  • is asymptotically exact, if superconvergence effects take place, and
  • provides information whether or not superconvergence has taken place.
This is a joint work in progress with Francesca Fierro (University of Milan).

(February 25) E. Darrigrand: Coupling of Fast Multipole Method and Microlocal Discretization for the Integral Equations of Electromagnetism - We are concerned with integral equations of scattering. In order to deal with the well-known high frequency problem, we suggest a coupling of two kind of methods that reduce the numerical complexity of iterative solution of these integral equations. The microlocal discretization method introduced by T. Abboud, J.-C. Ndlec and B. Zhou, enables one to reduce efficiently the size of the system considering an approximation of the phase function of the unknown. However, the method needs an expensive precalculation. We suggest the use of the fast multipole method introduced by V. Rokhlin, in order to speed up the precalculation. This work is an original application of the fast multipole method for acceleration of a microlocal discretization method within the new integral formulation written by B. Desprs. Numerical results obtained for Helmholtz's equation are very satisfying. For Maxwell's equations, they are also quite interesting.

(March 6) Alfio Quarteroni : Mathematics goes Sailing to the America's Cup - The application of numerical simulations to the design of sailing yachts is becoming more commonplace, particularly for the America's Cup. The Ecole Polytechnique Federale de Lausanne (Switzerland) has been the Official Scientific Advisor to the Alinghi Challenge for the America's Cup 2003. In the past eighteen months our research activity has been related to the application of advanced numerical methods for the solution of the mathematical equations governing the complex aerodynamic and hydrodynamic flows around the sailing yacht. In this talk I will report on the numerical studies that have been undertaken in three principal areas: hydrodynamic flow around the boat appendages (hull, keel, bulb and winglets), aerodynamic flow around the mast and sails, and the generation of waves on the water surface. The results obtained have been integrated into the design process, together with the results of more traditional numerical tools and experimentation,in order to optimize the overall boat performance. This problem is very challenging from the mathematical standpoint. The benefits to the design process as well as its limitations will be discussed. Practical matters, such as manpower and computational requirements, are also considered.

(March 11) A. Demlow: Properties of Mixed Finite Element Methods for Elliptic Problems - We consider two mixed finite element methods for a general second-order linear elliptic problem on a domain Omega in Rn. We present new L2 and localized pointwise Linfinity estimates which show that these two methods sometimes behave in substantially different ways in spite of the fact that they originate from the same differential equation. We also consider the effects of choosing various element spaces, chiefly the simplicial Raviart-Thomas and Brezzi-Douglas-Marini spaces, on the properties of the resulting approximations, and we show that the Brezzi-Douglas-Marini elements result in a suboptimal approximation in one of the methods.

(March 18) Andreas Prohl: Numerical Analysis of Nematic Liquid Crystals - Liquid Crystals share combined properties of crystals and liquids which is the reason for their attractivity in optical device applications.

In the 1960s, a continuum theory for nematics was proposed by J. Ericksen and F. Leslie to describe their dynamics. Analytical results were derived by R. Hardt, F.H. Lin, D. Kinderlehrer,.. starting from the late 80s (static case), and F.H. Lin & C. Liu over recent years (nonstationary case). Rigorous numerical analyses were initiated only very recently by the works of C. Liu & N. Walkington.

In the talk, we discuss efficient discretizations of the problem and their numerical analysis, with emphasis given to involved penalization parameters which control geometry of the microstructure evolved by the flow. We discuss our new approach, which uses tools from theory of harmonic mappings to verify first practically relevant error statements (`no crucial dependence on the penalization parameter'); by computational experiments, we verify that our results are sharp.

(April 1) C. T. Kelley: Design of Groundwater Remediation Systems with Sampling Method - In this talk we discuss several problems in optimal design of subsurface remediation and flow control systems. The objective functions and constraints are constructed from PDE solvers for the flow and species transport equations. We show how to implement problems using common production codes, how these codes and the underlying physics present obstacles to the optimization algorithms, and how sampling methods, in particular implicit filtering, can overcome these problems.

The objective functions and constraints in these problems are typically non-smooth in all the design parameters and discontinuous in some of them. We will discuss the physical reasons for these properties and how optimization methods can deal with them.

(April 15) Luminita Vese: PDE based image processing: an overview and some new results - In this talk, I will present three applications of nonlinear partial differential equations to image processing. A common technique is to derive the PDE models from energy minimization formulations. Here, the starting energy minimization problems will be: the Mumford-Shah problem for image segmentation, the total variation model of Rudin-Osher-Fatemi for image restoration, and the problem of harmonic maps. New results and new formulations based on these classical image analysis models will be presented, using the level set method of Osher-Sethian for curve evolution, and recent results of Meyer on oscillatory functions. The illustrated applications are:

  1. A multi-phase level set method for image segmentation and object detection
  2. Image decomposition models into cartoon plus texture
  3. A new formulation for processing of directional data
The mathematical formulations, as well as experimental results, will be presented.

(April 22) David Silvester: Error estimation and adaptivity for elliptic PDEs - In this talk we consider the design of robust and efficient methods for solving elliptic PDE problems. The essential ingredients are a low-order spatial approximation, and an a-posteriori error estimator that provides upper and lower bounds for the true spatial error. We illustrate our methodology in the context of convection-diffusion problems and steady incompressible flow problems.

Background and recent preprints

(April 29) Daniel Kessler: Phase-field models: a parameter hell - Phase-field models, the simplest of which is Allen-Cahn's problem, describe the evolution of a diffuse phase boundary, concentrated in a small region of size epsilon. They naturally call for adaptive mesh discretization techniques, which rely on error control based on a posteriori estimators. However, if Gronwall's lemma is used extensively, as is usually done in numerical analysis, the nonlinearity used to model phase separation leads to error estimates that grow exponentially as epsilon becomes small. When the interface region is thinnest, mesh adaption is needed most, but the error estimates justifying it are worst.

In this talk, we will show that a posteriori control of the error is in fact possible with only a polynomial dependence on the thickness of the interface region. The proof relies on a topological continuation argument, and the use of spectral estimates for Allen-Cahn's problem. As it is presented, the result is applicable to any conforming discretization technique that allows for a posteriori residual estimation. Residual estimators for an adaptive finite element scheme are then presented as an example.

Finally, we will mention some results for a phase-field model with anisotropy, where another paramater, the intensity of the anisotropy, is adding extra trouble to both theory and numerical experiments.

(May 6) Daniel B. Szyld: Convergence of Inexact Krylov Subspace Methods - Krylov subspace methods are possibly today's most widely use iterative methods for the solution of linear systems of algebraic equations in science and engineering. We give a short introduction to these methods and then provide a general framework for the understanding of Inexact Krylov subspace methods. In the inexact methods, the matrix-vector product at each step is not performed exactly. This framework allows us to explain the empirical results reported in the literature, where the exactness of the matrix-vector product is allowed to deteriorate as the Krylov subspace method progresses. A computable criterion is proposed to bound the inexactness of the matrix-vector multiplication in such a way as to maintain the convergence of the Krylov subspace method. The theory developed is applied to several problems. Numerical experiments are reported where the computable criteria are successfully applied. Furthermore, a theory is presented explaining the superlinear convergence of exact and inexact Krylov subspace methods.

(May 13) Mario Ohlberger: Robust a posteriroi error estimates for convection dominated weakly coupled parabolic systems - We consider a class of implicit finite volume schemes on unstructured grids to approximate solutions of convection dominated weakly coupled non-linear convection--diffusion--reaction systems. An a posteriori error estimate is proven. The L^1-error estimate obtained is robust in the diffusion coefficient, i.e. it applies in particular in the convection--dominated case and is even valid in the hyperbolic limit. Numerical experiments with an associated grid-adaptive algorithm are presented. Examples include environmental problems and combustion. From the numerical results it can be seen that the first order adaptive method is an adequate tool for non-linear convection with some self-sharpening effect. However, it is not convincing for linear advection problems because of its low order of convergence. To improve the method we introduce a higher order discretization of the convective part by MUSCL-type reconstruction. The improvement is demonstrated in several numerical examples.