(September 06) Prof. Jian-Guo Liu: Novel Finite Element Navier-Stokes Solvers without Inf-Sup Conditions - We will show that in bounded domains with no-slip boundary conditions, the Navier-Stokes pressure can be determined in a such way that it is strictly dominated by viscosity in an equivalent unconstrained formulation. As a consequence, in a general domain we can treat the Navier-Stokes equations as a perturbed vector diffusion equation, instead of as a perturbed Stokes system. We provide a simple proof of unconditional stability and convergence for discretization schemes that are implicit only in viscosity and explicit in both pressure and convection terms, requiring no solution of stationary Stokes systems or inf-sup conditions, particularly for corresponding fully discrete finite-element methods with $C^1$ elements for velocity and $C^0$ elements for pressure. It is important to note that we impose {\em no} inf-sup compatibility condition between the finite-element spaces for velocity and pressure. The inf-sup condition (also known as the Ladyzhenskaya-Babuska-Brezzi condition) has long been a central foundation for finite-element methods for all saddle-point problems including the stationary Stokes equation. Its beautiful theory is a masterpiece documented in many finite-element books. In the usual approach, the inf-sup condition serves to force the approximate solution to stay close to the divergence-free space where the Stokes operator is dissipative. However, due to the fully dissipative nature of the new unconstrained formulation, the finite-element spaces for velocity and pressure can be completely unrelated. This is a joint work with Bob Pego and Jie Liu.

(September 08) Prof. Christoph Schwab: Sparse Adaptive FEM for Multiple Scale Problems - Elliptic homogenization problems in a domain $\Omega \subset \R^d$ with $n+1$ separated scales are reduced to elliptic one-scale problems in dimension $(n+1)d$. These one-scale problems are discretized by a sparse tensor product finite element method (FEM). We prove that this sparse FEM has accuracy, work and memory requirement comparable to standard FEM for single scale problems in $\Omega$ while it gives numerical approximations of the correct homogenized limit as well as of all first order correctors, throughout the physical domain with performance independent of the physical problem's scale parameters.

Anisotropic Besov regularity in the scales and adaptive wavelet FEM will be addressed. Numerical examples for model diffusion problems with two and three scales will be given. The complexity of the method will be compared to that of the recently proposed hierarchical multiscale methods.

(September 13) Prof. Ohannes Karakashian: A Posteriori Error Estimates and Convergence of Adaptive Methods for a Discontinuous Galerkin Method for Elliptic Problems - A posteriori error estimates are derived for an interior penalty formulation for discontinuous Galerkin methods for elliptic problems. Residual type estimators as well as those based on the solution of local problems are formulated. The estimators are used to devise adaptive refinement strategies. It is shown that under certain conditions the algorithms are convergent with a guaranteed error reduction rate.

(September 20) Prof. Eberhard Baensch: Uniaxial, Extensional Flows in Liquid Bridges - Linear flow fields are commonly used for rheological studies, e.g. to measure the fluid viscosity or the deformation behavior of the whole sample or components in it. In this presentation we consider the possibility of generating homogeneous flows with a nearly constant strain rate.

This is achieved by stretching an almost cylindrical liquid bridge under microgravity. One key issue is the adjustability of the disk diameters, necessary for maintaining ideal boundary conditions. This method gives rise to much weaker end-effects than the commonly used method with unchangeable disks.

For the numerical simulation of the problem we use a finite element method with the following key ingredients: a variational formulation for the curvature of the free boundary, yielding an accurate, dimensionally-independent and simple-to-implement approximation for the curvature; a stable time discretization, semi-implicit with respect to the treatment of the curvature terms. This firstly allows one to choose the time step independently of the mesh size in contrast to common ``explicit'' treatments of the curvature terms, and secondly decouples the computation of the geometry and the flow field.

By systematic numerical simulations we could identify different regimes depending on $\Ca$, $\We$ and $\Re =\We / \Ca$. Regions of {\it capillary-dominated flow}, \hbox{$\Ca \ll 1$}, $\We \ll 1$, {\it viscous-dominated flow}, $\Ca > {\cal O}(0.1)$, $\Re < {\cal O}(0.1)$, and {\it inertia-dominated flow}, $\We > {\cal O}(0.1)$, $\Re >{\cal O}(0.1)$ have been detected, which exhibit mutually different bridge deformations during stretching.

(September 27) Chensong Zhang: A Posteriori Error Analysis for Parabolic Variational Inequalities in Finance - Motivated by the pricing American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain $\Omega\subset\mathbb{R}^d$ with an obstacle $\chi$. We formulate a fully discrete method by using piecewise linear finite elements in space and the backward Euler method in time. We define an a-posteriori error estimator and show that it gives an upper bound for the error in $L^2([0,T],H^1(\Omega)$. The error estimator is localized in the sense that the size of the elliptic residual is only relevant in the approximate non-contact region, and the approximability of the obstacle is only relevant in the approximate contact region. We also lower bound results for the local space error estimators in the non-contact region, and for the time error estimator. We present numerical results for $d=1,2$ which show that error estimator decays with the same rate $O(\tau + h)$ as the actual error when the space mesh size $h$ and the time step $\tau$ tend to zero. Also, the local error estimators capture the correct behavior of the errors in both the contact and the non-contact regions.  This is joint work with K-S. Moon, R.H. Nochetto and T. von Petersdorff.

(October 05) Prof. Chi-Wang Shu: Anti-diffusive High Order Weighted Essentially Non-Oscillatory Schemes for Sharpening Contact Discontinuities - In this talk we will first describe the general framework of high order weighted essentially non-oscillatory (WENO) finite difference schemes for solving hyperbolic conservation laws and in general convection dominated partial differential equations. We will then discuss our recent effort in designing anti-diffusive flux corrections for these high order WENO schemes. The objective is to obtain sharp resolution for contact discontinuities, close to the quality of discrete traveling waves which do not smear progressively for longer time, while maintaining high order accuracy in smooth regions and non-oscillatory property for discontinuities. Numerical examples for one and two space dimensional scalar problems and systems demonstrate the good quality of this flux correction. High order accuracy is maintained and contact discontinuities are sharpened significantly compared with the original WENO schemes on the same meshes. We will also report the extension of this technique to solve Hamilton-Jacobi equations to obtain sharp resolution for kinks, which are derivative discontinuities in the viscosity solutions of Hamilton-Jacobi equations. This is joint work with Zhengfu Xu.

(October 11) Prof. Alfred Schmidt: Adaptive Finite Element Methods for Allen-Cahn and Phase Field Problems - Phase field models, combining an Allen-Cahn equation for interface motion with a degenerate heat equation, are widely used to model phase transitions on a mesoscopic scale. Based on a posteriori error estimates, adaptive finite element methods for phase field problems are presented. For a double obstacle potential, the error indicators combine estimates for parabolic variational inequalities and degenerate diffusion problems. For a smooth potential, spectral estimates can be used to reduce the strong dependence of the estimator's constants on the inherent relaxation parameter.

The talk reports on (ongoing) joint works with Ricardo H. Nochetto, Zhiming Chen, and Daniel Kessler.

(October 18) Prof. Homer Walker: Globalization techniques for Newton-Krylov methods - A Newton-Krylov method is an implementation of Newton's method in which a Krylov subspace method is used to solve approximately the linear subproblems that determine Newton steps. To enhance robustness when good initial approximate solutions are not available, these methods are often "globalized", i.e., augmented with auxiliary procedures ("globalizations") that improve the likelihood of convergence from a poor starting point. In recent years, globalized Newton--Krylov methods have been used increasingly for the fully-coupled solution of large-scale CFD problems. In this talk, I will review several representative globizations, discuss their properties, and report on a numerical study aimed at evaluating their relative merits on large-scale 2D and 3D problems involving the steady-state Navier--Stokes equations. This is joint work with John Shadid and Roger Pawlowski at Sandia National Laboratories and Joseph Simonis at WPI.

(October 25) Cornelia Fermuller: Encounters of Vision with Scientific Computing - I will present two of our encounters of Scientific Computing with the vision system during the past few years. The first encounter gave rise to a theory that predicts many geometric optical illusions. The theory is based on estimation from limited amounts of data under noisy conditions. The second encounter was in the study of superresolution image reconstruction from sequences of low resolution images. Using the theory of wavelet filter banks we investigated the theoretical limits on the reconstruction and developed an efficient algorithm which is optimal when the noise model is unknown.

(November 1) Prof. Zdenek Strakos: Estimating the A-norm of the error in (Preconditioned) CG. - Estimation of the A-norm of the error in the conjugate gradient method is studied in many papers, reports and sections of books. In most of them the estimates are based on the fundamental relationship of the conjugate gradient and Lanczos methods to Gauss quadrature of the Riemann-Stieltjes integral defined by the spectral decomposition of the system matrix and components of the right hand side in the individual invariant subspaces.

After summarizing this fundamental relationship and recalling several possible ways of convergence evaluation, we present a simple estimate for the A-norm of the error in the preconditioned conjugate gradient method. We then describe results of numerical stability analysis and illustrate the effectivity and possible drawbacks of the proposed estimate on numerical experiments.

This represents joint work with Petr Tichy.

(November 8) Prof. Jerry Prince: Cortical Surface Alignment Using Optical Flow in an Eulerian Framework - How does one match locations on the human brain cortex of one person to that of another? A standard approach is deformable 3D-3D registration. This talk describes an alternative approach that focuses directly on the cortical surfaces and their geometric properties. Gyri and sulci are aligned in a multiresolution fashion using multiple geometric features computed on both cortices and computed in an Eulerian fashion. The result is an alignment of the cortices, which can then be used to apply a coordinate system to a given subject or to perform population analyses of function in a standardized coordinate system on the cortical surface.

(November 15) Prof. Marco Verani: A Finite Element Formulation for a Shape Optimization Problem - Several problems arising in Engineering can be formulated as shape optimization problems, i.e. the optimal design of an aorto-choronaric bypass, of a drug eluting stent or the optimal design of a rowing boat. In this talk I will show how it is possible to write a suitable FEM formulation for a simple Shape Optimization problem by coupling Shape Calculus tools and Finite Elements Methods for geometric flows.

(this is a joint project with G. Dogan, P. Morin and R. Nochetto)

(November 22) Prof. L. Ridgway Scott: The FEniCS Project to Automate Computational Mathematical Modeling - The objective of computational mathematical modeling is to make quantitative predictions of natural phenomena based on some type of mathematical description. Sophisticated models using partial differential equations can be found today in disciplines as diverse as economics, financial modeling, protein design, materials engineering, as well as more traditional areas. However, the development of software to implement these models remains costly and error-prone.

Numerous attempts have been made to develop problem-solving environments for partial differential equations (p.d.e's). Some of these have allowed (1) the specification of arbitrary p.d.e's, (2) the use of families of finite elements of arbitrary degree, (3) the incorporation of industrial-strength geometry engines, mesh generators, error estimators and adaptive mesh techniques, (4) the integration with parallel support tools, and (5) the use of multi-level and domain-decomposition solvers in addition to efficient sparse direct methods. However, no system has made use of all of these to the fullest extent possible, and there are important research issues related to doing so.

The FEniCS project is an international effort to study the fundamental issues related to automating the generation of software that can be used to approximate the solutions of p.d.e's and other similar systems. In addition to providing multiple end-user systems, FEniCS is developing what is called middle-ware to support such efforts. The concept of middle-ware goes beyond that of simple libraries, but the individual components of the middle-ware can be used independently and are being adopted by developers of other systems.

We will discuss the general outlines of the FEniCS project and then describe some specific efforts to generate efficient code for matrix assembly using a system called FErari. FErari can be thought of as part of a code generator which can be used for general finite elements and p.d.e's. We show that it is possible to have very efficient assembly for low-order methods, comparable to what is known for high-order methods using spectral-element technology.

(November 29) Prof. Alex Kiselev: Diffusion and Mixing in Fluid Flow - Enhacement of diffusion by advection is a classical subject that has been extensively studied by both physists and mathematicians. In this work, we consider enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow. Our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the flow amplitude is large enough. The necessary and sufficient condition on such flows is expressed naturally in terms of the spectral properties of the dynamical system associated with the flow. In particular, we find that weakly mixing flows always enhance the relaxation speed in this sense. The proofs are based on a new general criterion for the decay of the semigroup generated by a dissipative operator of certain form. They employ ideas from quantum dynamics, in particular the RAGE theorem describing evolution of a quantum state belonging to the continuous spectral subspace of the hamiltonian (and related to a theorem of Wiener on Fourier transforms of measures).

(December 06) Prof. Chun Liu: Macro-Micro Models in Viscoelastic Materials: An Energetic Variational Approach - I will describe an unified energetic variational framework for elastic complex fluids. It highlights the competition of the kinetic energy and the elastic energies, through the transport of the internal elastic variables. As applications, I will discuss some macro-micro scale coupling effects in these materials. Corresponding analytical and numerical issues will be addressed, in particular, I want to present some recent well-posedness results and a new moment closure algorithm that preserves the energy law of the original system.

(December 08) Prof. Willi Jaeger: Reactive Flow Through Porous Media - Consider flow, diffusion, transport and reactions through porous media consisting of a fluid and a solid phase. Assuming periodic media consisting of a fluid and a solid phase. Assuming periodic structures and a scale parameter \epsilon (measuring e.g. the pore size) and properly scaled model equations (micro-system) the scale limit \epsilon\to0 is studied under different assumptions. The aim is to derive macroscopic laws (macro-system) reducing the complexity of the model. The main mathematical problem consists in deriving the proper scale dependent estimates for the solutions to the given micro-system needed to apply the functional analytic tools of multi-scale convergence. Here, as usual nonlinear terms are posing obstacles in passing to the limit. Using localized coordinate systems and introducing macro- and microscopic variables, the arising compactness problems can be reduced essentially. In this seminar, the underlying concepts are introduced and applied to selected examples. Open problems arising in important applications will be discussed.

(December 09) Prof. Willi Jaeger: Multi-scale Modelling in Biosciences- Ion Transport Through Membranes - Modelling and simulation of processes in biosciences lead in general to complex model systems This complexity is caused by the system size, the arising nonlinearities, the range of scales involved, the stochastic nature of the processes and of the underlying geometry. It is a main challenge for Mathematics to reduce the complexity e.g. by analytic or numerical multi-scale methods. Already in setting up models these techniques are necessary to include information about the real processes on different scales and to link the corresponding models. The effect of microscale processes on the macro-scale behaviour has to be analysed. As an important example we consider the transport of ions through membranes. Mathematical modelling and simulation of ion concentrations inside and outside living cells, in their cyto-plasma and their nucleus, separated by membranes, are decisive for a better understanding of the bio-system "cell". Due to that fact that there is more and more information available about the processes on the micro-scale it is necessary to link model equations on micro-scale to a macroscopic description. It is important that model data can be computed using micro-scale information.

Here two domains are considered separated by a membrane perforated by channels placed in periodically distributed cells. The thickness of the membrane and the diameter of the cells are of order ?. The transport of ions is modelled by the Nernst-Planck equations, properly scaled in the channels. Charges fixed to the channels are included modelling the influence of the channels and the changes of its conformation. Effective laws for the ion transport trough membranes are derived performing an asymptotic analysis with respect to the scale parameter ?. The effective model consists in the Nernst-Planck equations on both sides of the membrane together with appropriate transmission conditions for the ion concentrations and the electric potential across the cell membrane. These conditions are determined solving micro-problems for cell problems in the membrane.

New methods of homogenization have to be developed and applied in order to deal with the nonlinear model equations and the reduction of the membrane to a two dimensional interface.

(December 13) Prof. Alexei Novikov: A homogenization approach to large-eddy simulation of incompressible fluids - In the development of large-eddy simulation one makes two primary assumptions. The first is that a turbulent flow can be categorized by a hierarchy of lengthscales. The second assumption states that the small scales have universal properties, characterized by, e.g. a spectral power law. This motivated a number of physical models that attempt to account for the presence of small scales by suitably modifying the corresponding partial differential equations (PDE), the Navier-Stokes equations. Homogenization theory addresses rigorously the issue of modification of PDE in the presence of small scales. The goal of this talk is to apply homogenization methods to LES modeling of fluid flows. We will start with the phenomenon of eddy viscosity in two-dimensions: in the presence of small-scale eddies the transport of large-scale vector quantities can be accompanied with depleted, and even ``negative" diffusion, when the Reynolds number is sufficiently large.