(September 3) Jean-Rodolphe Roche: Numerical Methods in Shape Optimization and Applications - Numerical methods of Newton-like type for shape optimization and related problems are presented. We consider applications where the cost function is a total energy and the state problem is a partial differential equation. We formulate a precise description of the first and second order shape derivatives and we give the technics to compute them. The numerical examples concerns the electromagnetic shaping in 2-d, 3-d and shape identification.

(September 17) Valeria Simoncini: The Nullspace-Free Eigenvalue problem and the Inexact Shift-and-Invert Lanczos Method - Given the large generalized symmetric eigenvalue problem A x =M x, with A semidefinite and M definite, we are interested in the approximation of the smallest nonzero eigenpairs, assuming that a sparse basis for the null space of A is available. A relevant feature is that the dimension of the null space is large, possibly up to one third of the problem dimension.

We analyze several algebraic formulations explored in recent years. We consider the inexact version of the Shift-and-invert Lanczos method, and we show that apparently different formulations provide the same approximation iterates, under some standard hypotheses.

(September 24) Yannis G. Kevrekidis: Equation-free multi scale computation: Enabling Microscopic Simulators to Perform System Level Tasks - I will present and discuss a framework for computer-aided multiscale analysis, which enables models at a "fine" (microscopic/stochastic) level of description to perform modeling tasks at a "coarse" (macroscopic, systems) level. These macroscopic modeling tasks, yielding information over long time and large space scales, are accomplished through appropriately initialized calls to the microscopic simulator for only short times and small spatial domains: "patches" in macroscopic space-time. Traditional modeling approaches first involve the derivation of macroscopic evolution equations (balances closed through constitutive relations). An arsenal of analytical and numerical techniques for the efficient solution of such evolution equations (usually Partial Differential Equations, PDEs) is then brought to bear on the problem. Our equation-free (EF) approach, introduced in PNAS (2000) when successful, can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form. We discuss how the mathematics-assisted development of a computational superstructure may enable alternative descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte Carlo (KMC) or Molecular Dynamics (MD) microscopic simulators, executed over relatively short time and space scales) to perform systems level tasks (integration over relatively large time and space scales, "coarse" bifurcation analysis, but also optimization and control tasks) directly. In effect, the procedure constitutes a systems identification based, "closure on demand" computational toolkit, bridging microscopic/stochastic simulation with traditional continuum scientific computation and numerical analysis. We illustrate these "numerical enabling technology" ideas through examples from chemical kinetics (LB, KMC), rheology (Brownian Dynamics), homogenization and the computation of "coarsely self-similar" solutions, and discuss various features, limitations and potential extensions of the approach.
Overview article

(October 8) Monique Dauge: Approximation of Maxwell singularities by nodal finite elements -

Near reentrant corners of a perfectly conducting boundary, electromagnetic fields have strong singularities that are not in H¹. The standard regularized variational formulation of the time-harmonic Maxwell equations, when discretized using nodal (C⁰) finite elements, leads to non-convergent Galerkin methods. The weighted regularization method [1,2] is a simple modification of the variational formulation that leads to convergent nodal finite element methods.

In its hp version, the WRM is particularly efficient. For 2D problems, exponential convergence can be shown. The method works well for 3D problems, too.

In the talk, some points from the proof of exponential convergence in 2D will be presented. The convergence behavior will be illustrated by the the results of computations in 2D and in 3D. The results in 3D indicate that good approximation requires an extremely strong geometric mesh refinement.

References:
[1] M. Costabel, M. Dauge: Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math Online publication DOI 10.1007/s002110100388 (2002)
[2] M. Costabel, M. Dauge, D. Martin, G. Vial: Weighted regularization of Maxwell equations - computations in curvilinear polygons. Proceedings of ENUMATH01, Springer 2002.

(October 15) Fabio Nobile: Some issues in the mathematical modelling and numerical simulation of the cardiovascular system - The simulation of the human cardiovascular system presents many challenging aspects in both modeling and set up of numerical tools. In this talk we will address two of them. The first one concerns the simulation of blood flow in a large artery when the deformation of the vessel wall is taken into account. We will present recent results on the so-called "Arbitrary Lagrangian Eulerian" (ALE) formulation, suitable to simulate fluid flow problems in domains with moving boundaries, and we will discuss the stability of some coupled fluid/structure algorithms. The second issue we will deal with concerns the possibility to achieve a global description of the circulatory system by combining models of different complexity and space dimension: what we called the "multiscale approach". This idea is motivated by the fact that local phenomena, as the presence of an atherosclerotic plaque or an implanted prosthesis, may have effects on the whole circulation, which should be predicted by the numerical simulation. We will give an overview on the different models available in the literature to describe blood circulation and we will present strategies to couple them in order to obtain a global model that accounts at the same time for local and systemic features.

(October 17) Gene H. Golub: Solution of Non-Symmetric, Real Positive Linear Systems - The methods we discuss use a Hermitian/skew-Hermitian splitting (HSS) iteration and its inexact variant, the inexact Hermitian/skew-Hermitian splitting (IHSS) iteration, which employs inner iteration processes at each step of the outer HSS iteration. Theoretical analyses show that the HSS method converges unconditionally to the unique solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the HSS iteration which is dependent solely on the spectrum of the Hermitian part. Numerical examples are presented to illustrate the effectiveness of both HSS and IHSS iterations. In addition, several important generalizations are presented.

(October 29) C. D. Levermore: The convergence of numerical transfer schemes in diffusive regimes - In regimes with small mean-free-paths the transfer equation with anisotropic boundary conditions has an asymptotic behavior that is governed by a diffusion equation and boundary conditions obtained through a matched asymptotic boundary layer analysis. A numerical scheme for solving this problem has a contribution to the truncation error that generally gives rise to a nonuniform consistency with the transfer equation as the mean-free-path vanishes, thus degrading its performance in diffusive regimes. We show that whenever certain numerical schemes have the correct diffusion limit, both in the interior and at the boundaries, their solutions converge to the solution of the transport equation uniformly in mean-free-path. The proof of this convergence is based on an asymptotic diffusion expansion and requires error estimates on a matched boundary layer approximation to the solution of the descrete equation.

(joint work with Shi Jin and Francois Golse)

(November 5) F. Seydou: An Inverse Problem for Anisotropic Objects - We discuss the problem of finding an anisotropic profile for the scattering of electromagnetic waves with fixed frequencies. This type of problem is one of the most important and interesting problems arising in mathematical physics and has applications in geophysics, technology, medicine and nondestructive testing. The difficulty in solving such problems is due to two unpleasant facts: They are nonlinear and, more seriously, they are ill-posed. In the previous works most algorithms were for TM and TE wave illuminations whereas much less was reported for the more complicated anisotropic case. We shall consider a simpler case of anisotropic medium. In Particular we shall assume the contrast to be orthotropic, but the problem is still much more complicated compared to the scalar case. We consider an optimization method to deal with the numerical approximation and discuss an iterative (the simplified Newton) method to try and solve the problem.

(November 12) Guowei Wei: A local spectral method for scientific computing - There has been an ongoing interest in computational methodology by numerous researchers in every field of science and engineering. Most effort has been centered in developing either global methods or local methods for solving a variety of problems. Global methods, such as Fourier or Chebyshev spectral methods, are usually highly accurate. But local methods, such as finite element and finite difference methods, are much more flexible for handling complex boundaries and geometries. We introduce a wavelet-like approach for achieving global methods' accuracy and local methods' flexibility in solving problems with complex boundaries and geometries. The mathematical foundation of the proposed method is the theory of distributions. Example applications are discussed to fluid dynamics, electromagnetics, solid mechanics and nonlinear waves.

(November 19) Zhiming Chen: A Convergent Adaptive Finite Element Algorithm with Reliable And Efficient Error Control for Linear Parabolic Problems - An efficient and reliable a posteriori error estimator is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator. A convergent adaptive algorithm with variable time-step sizes and space meshes is proposed and studied which, at each time step,, delays the mesh coarsening until the final iteration of the adaptive procedure, allowing only mesh and time-step size refinements before. The key ingredient in the convergence analysis is a new coarsening strategy. Numerical results are presented. This is a joint work with Feng Jia.

(November 26) Ming-Jun Lai: Multivariate Spline Method for Numerical Solution of PDE - I will explain how to use multivariate splines, piecewise polynomials of arbitrary degrees and smoothness over triangulation to numerically solve partial differential equations. The method does not require finite elements or locally supported basis functions and is very easy to implement. I will show how to use the stream function formulation for 2D Navier-Stokes equations and how to use the divergence free spline spaces for 3D Navier-Stokes equations. Some numerical experiments on standard cavity driven flows, backward facing step flows, flows around obstacles will be demonstrated.

(December 5) John Ball: Varying volume fractions - In elasticity models of materials which can undergo phase transformations involving a change of shape, different phases or variants of the material are associated with energy wells comprising corresponding sets of deformation gradients. Zero-energy microstructures can then be identified with gradient Young measures supported on these sets and the corresponding phase fractions are the masses of these measures restricted to each well.

The talk will describe a general result for gradient Young measures that allows one to vary the volume fractions of such microstructures, together with applications, open questions and related results.

(December 10) Susan Minkoff: Coupled Flow and Mechanics for Time-Lapse Seismic Modeling - Accurate prediction of oil and gas production in weak-formation reservoirs requires coupled flow and mechanical deformation modeling. We describe a loosely-coupled, staggered-in-time simulator which combines multiphase flow and advanced mechanics to model changes in reservoir properties (permeability and porosity) in compactible reservoirs during production. Numerical simulations indicate that these changes in flow parameters have a visible impact on time-dependent seismic models of the field despite large differences in scale between flow parameters and typical seismic wavelengths. (Joint work with Mike Stone, Sandia National Labs, Steve Bryant, Malgo Peszynska, Mary Wheeler, University of Texas at Austin)

(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.