Videos and presentation materials from other INI events are also available.
Event  When  Speaker  Title 

CPDW01 
20th January 2003 14:00 to 15:00 
R Rannacher 
The role of error control and adaptivity in numerical computation
Assessing accuracy in the numerical simulation of complex mathematical
models is one of the most natural but also most challanging problems
of numerical analysis. However, besides error control the aspect of
dimension reduction in the computation is gaining importance as the
complexity of models is growing faster than the capacity of computers.
During the last decade new approaches, largely based on mathematical
grounds, have been developed for the automatic, i.e. a posteriori,
adaptation of model complexity and discretization for various kinds
of problems. This talk will survey some of these developments including
examples from applications in Fluid Dynamics, Astrophysics and Chemistry.

CPDW01 
20th January 2003 15:30 to 16:30 
Error control and adaptivity for free boundary problems
This talk will assess the derivation of a posteriori error estimators for simple free boundary problems and emphasize novel techniques and open issues. The role of error control
in adaptivity will be exemplified via contact problems, curvature driven flows and crystal growth.


CPDW01 
20th January 2003 16:30 to 17:00 
J Ockendon 
Emerging areas for computational mathematics in industry
This talk will give an overview of the new industrial demands that have surfaced at recent Mathematics Study Groups and Faraday Workshops. The topics range from market behaviour to optimal lethality. Some specific examples will be listed, together with suggestions for new directions in computational research.

CPDW01 
20th January 2003 17:00 to 17:30 
Computational PDE's in industry Chair: Ockendon  
CPDW01 
21st January 2003 09:00 to 10:00 
O Pironneau 
Inverse fluid flow problems with discontinuities
There are many fluid flow problems with discontinuities in the data or in the flow. Among them three are quite important for applications:
Optimisation of these systems by standard gradient methods require the application of the techniques of the Calcul of Variations and an implicit asumption that a Taylor expansion exists with respect to the degrees of freedom of the problem. Take for example the flow in a transonic nozzle and the variation of the flow with respect to the inflow conditions; when these vary the shock moves and the derivative of the flow variables with respect to inflow conditions is a Dirac measure and so the Taylor expansion does not exists. By extending the calculus of variation via the theory of distribution it is possible to show however that the derivatives exists. But the result has serious numerical implications, in particular it favors the mixed finite element methods. We shall give numerical illustrations using the finite element method for an inverse problem for a Darcy flow, for the design of a transonic nozzle and for the design of a business supersonic airplane for sonic boom minimization. 
CPDW01 
21st January 2003 10:00 to 11:00 
Numerical diffusion induced grain boundary motion and methods for evolving interfaces
In this talk we describe applications involving curvature dependent interface motion and approaches to numerical approximation. In particular we consider grain boundary motion in thin metallic films induced by diffusion of atoms from a vapour. Three numerical approaches are introduced:1)Direct approximation of the sharp interface equations, 2)Phase field approximations and 3)Level set formulations.


CPDW01 
21st January 2003 11:30 to 12:30 
A Wheeler  Phasefield models 
CPDW01 
21st January 2003 14:00 to 15:00 
Atomistic simulations of diffusional processes with elastic interactions
Diffusion is a process that takes place at the atomic scale through the motion of defects, notably vacancies. In the first part of my talk I will show that the atomistic mechanisms of diffusion have consequences at the macroscale for kinetics and morphologies of phases produced by transformations. Even the mechanism of coarsening of second phases can be strongly influenced by the trapping of vacancies at interfaces.
In the second part of my talk I shall describe our most recent work on including longrange elastic interactions in atomistic models of diffusional processes. There are many instances where elastic interactions are known to have a profound effect on diffusional processes. In elastically anisotropic materials, spatial correlations of precipitates arise along elastically soft directions. Very small amounts of certain impurities can influence the kinetics of diffusional phase changes in a seemingly disproportionate manner. When atoms are deposited from the vapour onto surfaces islands may form to relieve the elastic strain energy in the system. These are all diffusional processes in which elastic interactions play a crucial role. The challenge for atomistic modelling is how to include very longrange effects in a simulation involving millions, even billions, of vacancy hops. We have developed new computational techniques to meet this challenge, which I shall describe and illustrate. This work was done in collaboration with Daniel R Mason, who was supported by EPSRC, and with Dr Robert Rudd of Lawrence Livermore National Laboratory. 

CPDW01 
21st January 2003 15:30 to 16:00 
Numerical approximation of surface diffusion
urface diffusion occurs in many mathematical models in material science.
In this talk we consider the numerical approximation of the
motion by surface diffusion (  the Laplacian of the curvature)
of a closed curve in the plane.
We review briefly the three standard numerical approximations in the literature.
Unfortunately, in contrast to the corresponding approximations of mean curvature flow, the underlying fourth order nature of the flow leads to a number of difficulties in the numerical analysis of these approaches. These mathematical challenges are highlighted in the talk. 

CPDW01 
21st January 2003 16:00 to 16:30 
Interface motion and phase field models Chair: Barrett, Elliott \& Wheeler  
CPDW01 
22nd January 2003 09:00 to 10:00 
Multiplescale modelling of materials using the quasicontinuum method
Atomistic and continuum methods alike are often confounded when faced with mesoscopic problems in which multiple scales operate simultaneously. In many cases, both the finite dimensions of the system as well as the microscopic atomicscale interactions contribute equally to the overall response. This makes modeling difficult since continuum tools appropriate to the larger scales are unaware of atomic detail and atomistic models are too computationally intensive to treat the system as a whole.
We present an alternative methodology referred to as the "quasicontiuum method" which draws upon the strengths of both approaches. The key idea is that of selective representation of atomic degrees of freedom. Instead of treating all atoms making up the system, a small relevant subset of atoms is selected to represent, by appropriate weighting, the energetics of the system as a whole. Based on their kinematic environment, the energies of individual "representative atoms" are computed either in nonlocal fashion in correspondence with straightforward atomistic methodology or within a local approximation as befitting a continuum model. The representation is of varying density with more atoms sampled in highly deformed regions (such as near defect cores) and correspondingly fewer in the less deformed regions further away and is adaptively updated as the deformation evolves. The method has been successfully applied to a number of atomicscale mechanics problems including nanoindentation into thin aluminum films, microcracking of nickel bicrystals, interactions of dislocations with grain boundaries in nickel, junction formation of dislocations in aluminum, crossslip and jogdrag of screw dislocations in copper, stressinduced phase transformations in silicon due to nanoindentation, polarization switching in ferroelectric leadtitanate and deformation twinning at aluminum crack tips. An overview of the methodology and selected examples from these applications will be presented. 

CPDW01 
22nd January 2003 10:00 to 11:00 
B Engquist 
Heterogeneous multiscale methods
The heterogeneous multiscale method is a framework for numerical approximations of problems with very different active scales. The original problem is discretized on the macroscale and the missing data is supplied by microscale simulations on subdomains. The use of subdomains is critical in order to reduce the overall computational
complexity. Examples will be given from homogenization of partial differential equations, Brownian motion and molecular dynamics. The emerging theory will also be discussed.

CPDW01 
22nd January 2003 11:30 to 12:30 
Molecular models of viscoelastic flows
The NonNewtonian flow of polymer melts and solutions in complex geometries generates very novel fluid mechanics: noninertial turbulence, elastic vortex growth, elastic freesurface swell. Moreover, even the qualitative nature of these features is now known to be sensitive to the detailed moledcular architecture of the polymers themselves. Simple phenomenology is therefore unable to produce realistic constitutive equations (computational strategies for the stress that, together with the Stokes equations of flow generate the set of PDEs for the viscoelastic flow). Instead, molecular physics has more recently suggested forms of constitutive equation for linear and branched molecules that contain more than one relaxation time, and sometimes other structural variables that are needed for the equations of motion to close.
Computing with these equations is also a challenge: two classes of solver have emerged  Eulerian and Lagrangian. The latter approach lends itself much more readily to the physics, but us far less well studied than the fixedgrid case. Properties of the stablility and convergence of Lagrangian methods over Eulerian represent a major current challenge. 

CPDW01 
22nd January 2003 14:00 to 15:00 
Analysis and applications of the heterogeneous multiscale method?
The heterogeneous multiscale method (HMM) is a general methodology for the efficient numerical computation of problems with multiscales and multiphysics. The method
relies on an efficient coupling between the macroscopic and microscopic models. In case when the macroscopic model is not explicitly available or invalid, the microscopic solver is used to supply the necessary data for the macroscopic model. Scale separation can be exploited to considerably reduce the complexity of the microscopic solver. Besides unifying several existing multiscale methods such as the quasicontinuum method and kinetic schemes in gas dynamics, HMM also provides a methodology for designing new methods
for a large variety of multiscale problems. We will present this framework as well as applications to homogenization, porous medium flows, molecular dynamics and interface dynamics. We will also present a framework for analyzing the stability and accuracy of HMM.


CPDW01 
22nd January 2003 15:30 to 16:00 
Adaptive finite element methods for hyperbolic problems
The purpose of the lecture is to present an overview of recent developments in the area of residualbased a posteriori error estimation for finite element approximations of partial differential equations, and the implementation of the a posteriori error bounds into adaptive finite element algorithms. We shall be particularly concerned with h and
hpadaptive algorithms for firstorder hyperbolic systems and secondorder partial differential equations of mixed elliptichyperbolicparabolic type. The theoretical results will be illustrated by numerical experiments. This is joint work with Paul Houston, University of Leicester, UK.


CPDW01 
22nd January 2003 16:00 to 16:30 
Adaptive finite element methods for partial differential equations Chair: Suli  
CPDW01 
23rd January 2003 10:00 to 11:00 
R Glowinski 
Numerical methods for the solution of a system of eikonal equations with Dirichlet boundary conditions
In this presentation, we discuss the numerical solution of a system of eikonal equations with Dirichlet boundary condition. Since the problem under consideration has infinitely many solutions, we look for those solutions which are nonnegative and maximal (or nearly
maximal) in the $L^1$norm. The computational methodology combines penalty, biharmonic regularisation, operator splitting, and finite element approximation. Its practical implementation requires essentially the solution of cubic equations in one variable and of discrete linear elliptic problems of the Poisson and Helmholtz type. As expected, when the
spatial domain is a square whose sides are parallel to the coordinate axes, and when the Dirichlet data vanishes at the boundary, the computed solutions show a fractal behavior near the boundary, and particularly, close to the corners.

CPDW01 
23rd January 2003 11:30 to 12:30 
T Chan 
Variational PDE techniques in wavelet transforms \& image compression
Standard wavelet linear approximations (truncating high frequency
coefficients) generate oscillations (Gibbs' phenomenon) near singularities
in piecewise smooth functions. Nonlinear and data dependent methods are
often used to overcome this problem. Recently, a new research direction has
emerged, which introduces partial differential equation (PDE) and
variational techniques (including techniques developed in computational
fluid dynamics (CFD)) into wavelet transforms for the same purpose.
Two different approaches have been used. One is to use PDE ideas to directly change wavelet transform algorithms so as to generate wavelet coefficients which can avoid oscillations in reconstructions when the high frequency coefficients are truncated. The other one is to stay with standard wavelet transforms and use variational PDE techniques to modify the coefficients in the truncation process so that the oscillations are reduced in the reconstruction processes. In this talk, I will present an overview of our work on both approaches of this direction. The first part will be on an adaptive ENO wavelet transform designed by using ideas from Essentially NonOscillatory (ENO) schemes for numerical shock capturing. ENOwavelet transforms retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining arbitrary high order accuracy uniformly and having a multiresolution framework and fast algorithms, all without any edge artifacts. We have also shown the stability of the ENOwavelet transform and obtained a rigorous approximation error bound which shows that the error in the ENOwavelet approximation depends only on the size of the derivative of the function away from the discontinuities. Applications to image compression will be briefly mentioned. The second part of the talk is on using a variational framework, in particular the minimization of total variation (TV), to select and modify the retained standard wavelet coefficients so that the reconstructed images have fewer oscillations near edges. 
CPDW01 
23rd January 2003 14:00 to 14:20 
Preconditioning the incompressible Navier Stokes equations  
CPDW01 
23rd January 2003 14:20 to 14:40 
Direct numerical simulation of dropletgas systems for power generation applications  
CPDW01 
23rd January 2003 14:40 to 15:00 
Variational techniques for micromagnetic simulations  
CPDW01 
23rd January 2003 16:00 to 16:20 
Integral equation methods for direct and inverse scattering by unbounded surfaces  
CPDW01 
23rd January 2003 16:20 to 18:00 
EPSRC Programme managers in mathematics \& engineering \& high performance computing  
CPDW01 
24th January 2003 09:00 to 10:00 
A Quarteroni 
Numerical modelling of the cardiovascular system
The use of numerical simulation in the study of the cardiovascular system (and its inherent pathologies) has greatly increased in the past few years. Blood flow interacts mechanically and chemically with vessel walls producing a complex fluidstructure interaction problem, which is practically impossible to simulate in its entirety.
Several reduced models have been developed which may give a reasonable approximation of averaged quantities, such as mean flow rate and pressure, in different sections of the cardiovascular system. They are, however, unable to provide the details often needed for understanding a local behavior, such as, e.g., the effect on the shear stress distribution due a modification in the blood flow consequent to a partial stenosis. The derivation of these heterogeneous models, and their coupling, will be presented together with schemes for their numerical solution. These techniques may be extended by including models for chemical transport. Several numerical results on cases of real life will be presented. 
CPDW01 
24th January 2003 10:00 to 11:00 
P Monk 
Inverse electromagnetic scattering
In this talk we shall consider the inverse scattering problem of determining the shape of an object from electromagnetic data. In particular we are motivated by radar applications in which we would like a rapid solution from incomplete data. We shall discuss a model
problem using far field data at a single frequency in the resonance region where we encounter challenges due to the illposed nature of the problem.
The classical method for solving this problem, which has been used with great success, is to linearize the mapping from the boundary to the the far field pattern. However, the resulting Synthetic Aperture Radar technique has been criticized because it is based on an inaccurate model. This has lead researchers to consider the full nonlinear inverse problem. After a brief discussion of optimization approaches to this problem (which are generally very computationally intensive), we shall discuss recent developments in inverse electromagnetic scattering based on linear sampling methods. We end with some comments on open problems and challenges for the future. 
CPDW01 
24th January 2003 11:30 to 12:00 
AR Davies 
PDE's, food rheology, and processing
Liquidlike food is essentially a complex mixture of natural polymers in an aqueous solvent of disperse biological material. Very few liquids foods are Newtonian fluids satisfying the NavierStokes equations. For example, creamy soups, ketchup and yoghurt are shearthinning; concentrated starches are shearthickeining; dough is viscoelastic, while paté and processed cheese are effectively soft solids. The challenges in modelling the processing of food in the liquid state are not confined, however, to nonNewtonian considerations: in addition to modelling the rheology of the food one must also consider parameter identification; temperature dependence and thermal boundary conditions; slip and lubrication; complexity of the flow and stress concentrations resulting from the shape of the manufacturing apparatus. High pressures encountered in injection moulding, and cavitation are also issues which must be addressed in certain situations. This lecture will illustrate some of these challenges, show how some are being addressed, and also list a number of open mathematical problems.

CPDW01 
24th January 2003 12:00 to 12:30 
PDE's in food industry Chair: Davies  
CPDW01 
24th January 2003 14:00 to 14:45 
Mesh quality in computational engineering
Fundamental to finite element and finite difference analysis is a spatial discretisation of the domain i.e. the construction of a mesh. For simple geometries, this is not an issue. However, when the geometry is complicated, the construction of an appropriate mesh can be a major challenge. In fact, the automatic construction of a high quality mesh is often cited as the major problem preventing the more widespread use of computational analysis via finite element and finite difference analysis.
Although the stateoftheart in mesh generation has advanced in the last 10 years, there are still major issues that need to be addressed. Using examples from computational fluid dynamics and computational electromagnetics, the presentation will highlight some new challenges and opportunities for significant advances. Some recent developments using new and innovative ideas will be presented. . 

CPDW01 
24th January 2003 14:45 to 15:00 
Mesh generation Chair: Weatherill  
CPDW01 
24th January 2003 15:30 to 16:00 
M Giles 
Challenges in scientific and engineering computation
In the talk I will aim to be provocative, to stimulate discussion in the final session. I will mention some of the work I am doing with adjoint methods, but mostly I will present what I see as challenges in the areas of scientific and engineering computation, not all of which are mathematical. As the final speaker, I will also attempt to draw on the presentations of all of the previous speakers in the week.

CPDW01 
24th January 2003 16:00 to 16:30 
Engineering computations Chair: Giles  
CPDW01 
24th January 2003 16:30 to 17:00 
Unknown  
CPD 
5th February 2003 15:30 to 16:30 
D Silvester  Fast blackbox preconditioners for elliptic PDE problems 
CPD 
5th February 2003 17:00 to 18:00 
Time discretisation of an evolution equation via Laplace transforms  
CPD 
6th February 2003 15:00 to 16:00 
C Schwab  Sparse finite elements for PDEs with stochastic data 
CPDW02 
13th February 2003 10:00 to 11:00 
Micromagnetism  some current challenges faced by experimentalists
Small magnetic structures play a crucial role in modern storage devices and sensors. To achieve ever improving performance, magnetic films with superior properties and smaller (and frequently more densely packed) magnetic elements are required. When experimenting with new magnetic systems the ideal would be to have a description of the magnetisation distribution throughout the 3dimensional magnetic system and to understand how it evolves as a function of vector field, temperature and time. Resolution as close as possible to the atomic level, and in any event better than characteristic magnetic length scales, is also highly desirable. Despite the rapid development of existing and the introduction of new magnetic microscopy techniques over the last decade, we still fall far short of these goals. In this talk, I will give examples of some recent imaging investigations with equal emphasis on the information that they do and do not provide.
Among the topics to be discussed will be the extent to which the properties of notionally identical magnetic elements differ as a result of small differences in their physical microstructure. This can be quite complex in elements comprising a single magnetic layer but is inevitably much more so in multilayers where the detailed coupling between the layers themselves remains imperfectly understood. Further challenges arise when the elements are packed closely together as interactions between them become important. Other significant topics are the effect of thermal processes leading to property changes with time, all other parameters remaining constant. Studies of this nature relate, for example, to stability and hence the suitability of particular structures for storage purposes. At the other end of the time spectrum is the phenomenon of ultrafast switching where it is difficult to simultaneously meet the requirements of high spatial and temporal resolution. I will discuss how combining information gleaned from experiment and micromagnetic modelling can help to advance the subject. 

CPDW02 
13th February 2003 11:30 to 12:30 
Time and length scales in computational microgenetics
Micromagnetics is a continuum theory to describe magnetization processes on a length scale which is large enough to replace the atomic spins by a continuous magnetization vector and small enough to resolve the magnetization transition inside a domain wall. The characteristic length scale, the domain wall witdh, is in the order of five to ten nanometers,
The size of the basic structural units of magnetic devices like sensor or storage elements may extend towards micrometers. The time scale range from subnanosecond regime for fast precessional switching to years for thermally activated magnetization reversal. The wide range of time and length scales involved in micromagnetic simulations is a challenge for effective computational tools. In the talk numerical methods for the solution of the LLG equation and the simulation of thermal stability will be reviewed. In particular, the role of the physical microstructure on hysteresis properties will be discussed, and numerical techniques to treat microstructural effects will be proposed.


CPDW02 
13th February 2003 14:00 to 15:00 
A Prohl 
Numerical Analysis of stationary and nonstationary micromagnetism
Magnetic patterns of ferromagnetic materials are described by the wellaccepted model of Landau and Lifshitz. Over the last years, different strategies have been developed to tackle the related nonconvex problem: direct minimization, convexification, and relaxation by using Young measures. Nonstationary ferromagnetic effects are considered by the extended model of Landau, Lifshitz and Gilbert (LLG).
In the talk, we survey numerical analysis for solving the stationary problem. In the second part, we discuss proper projection/penalization strategies to discretize (LLG).

CPDW02 
13th February 2003 15:30 to 16:15 
Weak and regular solutions for LandauLifschitz equations: existence and asymptotic behaviour
In this talk, we describe existence results for weak and regular solutions of LandauLifschitz equations. Furthermore, we present two types of asymptotic behaviour : for large time, we study the $\omega$limit set of the weak solutions. On the other hand, we perform an asymptotic expansion of the regular solutions when the exchange coefficient goes to zero : it appears a boundary layer described with a BKW method.


CPDW02 
13th February 2003 16:15 to 17:00 
P Monk  An eddy currentmicromagnetic model with application to disk write heads 
CPDW02 
14th February 2003 10:00 to 11:00 
Using PDE's to develop the application of superconductors  
CPDW02 
14th February 2003 11:30 to 12:30 
Macroscopic models of superconductivity  
CPDW02 
14th February 2003 14:00 to 15:00 
Q Du 
Quantized vortices: from GinzburgLandau to GrossPitaevskii
Quantized vortex, a signature of superfluidity, has been studied for more than fifty years
in the context of superconductivity and in recent years, it has attracted much attention in
the study of the BoseEinstein condensates.
In this talk, we will discuss some questions related to quantized vortices based on the GinzburgLandau models for superconductivity and the GrossPitaevskii equations for the BoseEinstein condensation. Some strikingly similar properties will be illustrated. Both mathematical results concerning these equations and the computational challenges will be presented. 
CPDW02 
14th February 2003 15:30 to 16:15 
V Styles 
A finite element approximation of a variational formulation of Bean's model for superconductivity
We introduce phase field and sharp interface models for bidirectional diffusion induced grain boundary motion with triple junctions. We present numerical discretizations of the models together with some computational simulations.

CPDW02 
14th February 2003 16:15 to 17:00 
Solution of criticalstate problems in superconductivity  
CPD 
19th February 2003 15:30 to 16:30 
FEMBEM coupling for quasistatic electromagnetism  
CPDW08 
4th March 2003 10:25 to 10:30 
Introduction  
CPDW08 
4th March 2003 10:30 to 11:15 
Bread crusting  
CPDW08 
4th March 2003 11:15 to 12:00 
Scraped surface heat exchangers  
CPDW08 
4th March 2003 12:00 to 12:45 
Microwaving  
CPDW08 
4th March 2003 14:00 to 14:45 
RD Davies  Computational algorithms overview 
CPDW08 
4th March 2003 14:45 to 15:30 
Chaired discussion of critical computational issues in the three projects presented  
CPDW08 
4th March 2003 16:00 to 17:00 
Open Forum: A discussion of food industry problems involving mathematics and computation  
CPDW08 
4th March 2003 17:00 to 00:00 
Summing up and close  
CPDW08 
5th March 2003 10:25 to 10:30 
Introduction  
CPDW08 
5th March 2003 10:30 to 10:50 
Computational algorithms for inertiadominated free surface flows  
CPDW08 
5th March 2003 10:50 to 11:35 
L Davenport & A Ruddle & C Tully  Electromagnetic compatibility 
CPDW08 
5th March 2003 11:35 to 12:20 
Droplet impact on water layers  
CPDW08 
5th March 2003 12:20 to 13:00 
J Curtis & J Ockendon  Shaped charge mechanics 
CPDW08 
5th March 2003 14:00 to 14:30 
C Johnson  Finite element software 
CPDW08 
5th March 2003 14:30 to 15:00 
P Monk  Computational algorithms in electromagnetics 
CPDW08 
5th March 2003 15:00 to 15:30 
Chaired discussion of critical computational issues in the three projects presented  
CPDW08 
5th March 2003 16:00 to 17:00 
Open Forum: A discussion of industrial electromagnetic and violent mechanical problems involving mathematics and computation  
CPDW08 
5th March 2003 17:00 to 00:00 
Summing up and close  
CPD 
12th March 2003 16:00 to 17:00 
An adaptive finite element method for minimal surfaces  
CPD 
19th March 2003 16:00 to 17:00 
A generalised discretisation concept for optimal control with pdes  
CPD 
26th March 2003 16:00 to 17:00 
Convergent adaptive finite elements for the plaplacian  
CPDW03 
7th April 2003 09:55 to 10:00 
Opening remarks  
CPDW03 
7th April 2003 10:00 to 11:00 
F Brezzi 
Pseudo residual free bubbles and subgrids
Residual free bubbles proved to be a powerful technique to deal with subscale
phenomena. In particular they are known to provide optimal stabilizing
effects in many circumstances, where traditional methods can exhibit wide
oscillations or checkerboard modes.
As it is well known, however, residual free bubbles require the solution of a partial differential equation (of the same nature of the original one). This can only be done in some approximate way. These approximate solutions are often called ``pseudo residual free bubbles" In a certain number of problems, the accuracy needed in this element by element resolution is not very high, and can be dealt with by using a very poor (but suitably chosen!) subgrid, consisting in just a few additional nodes. This occurs for instance when we want to stabilize advectiondiffusion or advectionreactiondiffusion problems. In these cases we can just consider that we are adding a suitable subgrid and then solving plain Galerkin on the augmented grid (=original grid + subgrid) In other problems there is a need to combine in an appropriate way the element subgrid with a boundary subgrid. In the boundary subgrid one can also consider non polynomial functions, such as suitable exponential or numerical solution of suitable auxiliary problems solved in small regions with a finer grid. The lecture will present an overview of these issues and on latest developments in these directions. 
CPDW03 
7th April 2003 11:30 to 12:30 
Analysis of multiscale methods
We will discuss the numerical analysis of multiscale methodsfor several class of multiphysics problems


CPDW03 
7th April 2003 14:00 to 15:00 
T Hou 
Multiscale computation for flow through heterogeneous media
Many problems of fundamental and practical importance contain
multiple scale solutions. Composite materials, flow and transport in
porous media, and turbulent flow are examples of this type. Direct
numerical simulations of these multiscale problems are extremely
difficult due to the range of length scales in the underlying physical
problems. Here, we introduce a dynamic multiscale method for computing
nonlinear partial differential equations with multiscale solutions.
The main idea is to construct semianalytic multiscale solutions
local in space and time, and use them to construct the coarse grid
approximation to the global multiscale solution. Such approach overcomes
the common difficulty associated with the memory effect in deriving the
global averaged equations for incompressible flows with multiscale solutions.
It provides an effective multiscale numerical method for computing twophase
flow and incompressible Euler and NavierStokes equations with multiscale
solutions. In a related effort, we introduce a new class of numerical
methods to solve the stochastically forced NavierStokes equations. We will
demonstrate that our numerical method can be used to compute accurately
high order statitstical quantites more efficiently than the traditional
MonteCarlo method.

CPDW03 
7th April 2003 15:30 to 16:30 
Adaptive FEM for saddle point problems: design \& convergence
The lack of monotonicity in saddle point problems is a
fundamental impediment to prove convergence of direct adaptive
finite element methods (AFEM). We propose an alternative
consisting of two nested iterations, the outer iteration being an
Uzawa algorithm to update the scalar variable and the inner
iteration being an elliptic AFEM for the vector variable. We show
linear convergence in terms of the outer iteration counter
provided the elliptic AFEM guarantees an error reduction rate
together with a reduction rate of data oscillation (information
missed by the underlying averaging process). We apply this idea
to the Stokes system without relying on the discrete infsup
condition; unstable elements are thus nonlinearly stabilized.
We also deal with an augmented Lagrangian formulation for the
RaviartThomas mixed finite elements, and discuss error control
and error reduction rate for the corresponding elliptic operator.
We finally assess complexity of the elliptic AFEM, and provide
consistent computational evidence that the resulting meshes are
quasioptimal. This work is joint with E. Baensch, M. Cascon,
P. Morin, and K.G. Siebert.


CPDW03 
7th April 2003 16:30 to 17:00 
K Siebert 
Convergence of adaptive finite element methods
Adaptive finite element methods (FEM) have been widely used in
applications for over 20 years now. In practice, they converge
starting from coarse grids, although no mathematical theory has been
able to prove this assertion. Ensuring an error reduction rate
based on a posteriori error estimators, together with a reduction
rate of data oscillation (information missed by the underlying
averaging process), we construct a simple and efficient adaptive FEM
for elliptic partial differential equations. We prove that this
algorithm converges with linear rate without any preliminary mesh
adaptation nor explicit knowledge of constants. Any prescribed error
tolerance is thus achieved in a finite number of steps. A number of
numerical experiments in two and three dimensions yield
quasioptimal meshes along with a competitive performance.

CPDW03 
7th April 2003 17:00 to 17:45 
Informal time for researchers from various disciplines to exchange ideas \& explore currently unsolved problems  
CPDW03 
8th April 2003 09:00 to 10:00 
R Rannacher 
Adaptive discretisation of optimisation problems with PDE
We present a systematic approach to error control and mesh
adaptivity in the numerical solution of optimization
problems with PDE constraints. By the Lagrangian formalism
the optimization problem is reformulated as a saddlepoint
boundary value problem which is discretized by a Galerkin
finite element method. The accuracy of this discretization
is controlled by residualbased a posteriori error
estimates. The main features of this method will be
illustrated by examples from optimal control of fluids
and parameter estimation.

CPDW03 
8th April 2003 10:00 to 11:00 
Coordinated discussion session: Adaptivity  
CPDW03 
8th April 2003 11:30 to 12:30 
C Johnson 
Adaptive computational modelling of reactiondiffusion in laminar and turbulent incompressible flow
We present a framework for adaptive computational modeling
with application to reactiondiffusion in laminar and incompressible flow.
We estimate using duality the total computational error in different
outputs a posteriori, with contributions from both discretization in
space/time and subgrid modeling of unresolved scales. We consider subgrid
models based on extrapolation or local resolution of subgrid scales. We
present computational results for laminar and turbulent Couette flow.

CPDW03 
8th April 2003 14:00 to 15:00 
C Schwab  Finite elements for homogenisation problems 
CPDW03 
8th April 2003 15:30 to 16:00 
hpadaptive discontinuous Galerkin methods with interior penalty for degenerate elliptic partial differential equations
We consider the a posteriori and a priori error
analysis of the hpversion of the discontinuous Galerkin finite
element method with interior penalty for approximating
secondorder partial differential equations with nonnegative
characteristic form.
In particular, we discuss the question of error estimation
for certain linear target
functionals of the solution of practical interest;
relevant examples include the local
mean value of the field or its flux through the outflow boundary
of the computational domain $\Omega$, and the evaluation
of the solution at a given point in $\Omega$.
Our a posteriori error bounds stem from a
duality argument and include
computable residual terms multiplied
by local weights involving the solution of a certain
dual or adjoint problem.
Guided by our a posteriori error analysis,
we design and implement an adaptive finite element algorithm
to ensure reliable and efficient control of the error in the
computed functional with respect to a userdefined tolerance. A key
question in hpadaptive algorithms is how to automatically
decide when to hrefine/derefine and when to prefine/derefine.
To this end, we construct an appropriate adaptive strategy
based on estimating the
local Sobolev regularities of the primal and dual solutions.
The performance of the resulting hprefinement algorithm is
demonstrated through a series
of numerical experiments. In particular, we demonstrate the
superiority of using hpadaptive mesh refinement with the
traditional hrefinement method, where the degree of the
approximating polynomial is kept fixed at some low value.


CPDW03 
8th April 2003 16:00 to 16:30 
J Mackenzie 
Adaptive moving mesh techniques for moving boundary value problems
There is considerable engineering and scientific interest in the
solution of phase change problems. In this talk we consider the
use of adaptive moving mesh methods to efficiently model these
moving boundary value problems. These methods are based on a
redistribution of mesh nodes in time while normally keeping the
number and connectivity of mesh elements fixed thus leading to less
complex algorithms. We consider the derivation of the moving mesh
partial differential equations (MMPDEs) and appropriate adaptivity criteria.
Numerical simulations of the phasefield equations and buoyancy
induced heat transfer problems will be shown to demonstrate the
potential of this approach which ideally should be combined with
other forms of solution adaptivity.

CPDW03 
8th April 2003 16:30 to 17:00 
An optimal adaptive finite element method
We present an adaptive finite element method for solving second order elliptic equations which is (quasi)optimal in the following sense: If the solution is such that for some s>0, the errors in energy norm of the best continuous piecewise linear approximations subordinate to any partition with N triangles are O(N^{s}), then given an eps>0, the adaptive method produces an approximation with an error less than eps subordinate to a partition with O(eps^{1/s}) triangles, taking only O(eps^{1/s}) operations. Our method is based on ideas from [Binev, Dahmen and DeVore '02], who added a coarsening routine to the method from [Morin, Nochetto and Siebert '00]. Differences are that we employ nonconforming partitions, our coarsening routine is based on a transformation to a wavelet basis, all our results are valid uniformly in the size of possible jumps of the diffusion coefficients, and that we allow more general righthand sides. All tolerances in our adaptive method depend on a posteriori estimate of the current error instead an a priori one, which can be expected to give quantitative advantages.


CPDW03 
8th April 2003 17:00 to 18:30 
Informal time for researchers from various disciplines to exchange ideas \& explore currently unsolved problems  
CPDW03 
9th April 2003 09:00 to 10:00 
R DeVore 
Tools for adaptive finite element methods
We will discuss two results that have proven to be important in the
analysis of convergence rates for adaptive finite element methods.
The first is how to bound the number of additional subdivisions needed
to remove hanging nodes in newest vertex bisection. Such a bound is
necessary to control complexity bounds in adaptive methods. The second
result is how to generate a near best adaptive approximation to a given
function in linear time. This result has been used in coarsening
routines to keep control of the number of cells in adaptively generated
triangular partitions. This is joint work with Peter Binev and Wolfgang Dahmen.

CPDW03 
9th April 2003 10:00 to 11:00 
Coordinated discussion session: Wavelets and multiresolution for PDEs  
CPDW03 
9th April 2003 11:30 to 12:30 
A Cohen 
Multiscale adaptive schemes for hyperbolic equations: some convergence results
we shall discuss the convergence analysis of a class of adaptive
schemes for evolution equations. These schemes combine adaptive mesh
refinements with wavelets based on a framework introduced by Ami Harten.
The analysis will mainly focus on the hyperbolic case.

CPDW03 
9th April 2003 14:00 to 15:00 
W Dahmen 
Adaptive multiscale application of operators
This talk is concerned with the adaptive application of
(linear and nonlinear)
operators in wavelet coordinates. There are two principal steps, namely
first the reliable prediction
of significant wavelet coefficients of the result of such applications
from
those of the input, and second the accurate and efficient computation of
the significant output coefficients.
Some of the main conceptual ingredients of such schemes are discussed.
Moreover, we indicate
how this leads to adaptive solution schemes for variational problems
that can be shown to
have asymptotically optimal complexity. This is joint work with A. Cohen
and R. DeVore.

CPDW03 
9th April 2003 15:30 to 16:00 
Adaptive wavelet methods for control problems with PDE constraints
For the fast numerical solution of a control problem governed by a stationary PDE
with distributed or Neumann boundary control, an adaptive algorithm is proposed
based on wavelets.
A quadratic cost functional involving natural norms of the state and the control
is to be minimized subject to constraints in weak form.
Placing the problem into the framework of (biorthogonal) wavelets allows to
formulate the functional and the constraints equivalently in terms
of sequence norms of wavelet expansion coefficients and constraints in form of an
automorphism. The resulting first order necessary conditions are then derived as
a (still infinite) weakly coupled system of equations.
Once this system is obtained, the machinery developed by Cohen, Dahmen and DeVore
can be employed for the design of an adaptive method which can be interpreted as an inexact
gradient method. In each iteration step the primal and the dual system needs to be
solved up to a prescribed accuracy. In particular, we show that the adaptive algorithm
is asymptotically optimal, that is, the convergence rate achieved
for computing the solution up to the target tolerance is asymptotically
the same as given by the waveletbest Nterm
approximation of the solution, and the total computational work is proportional
to the number of unknowns.


CPDW03 
9th April 2003 16:00 to 16:30 
K Urban 
Adaptive convex optimisation: convergence results
We consider the minimization of a convex functional on reflexive Banach
spaces. We introduce an infinitedimensional (exact) algorithm of
steepest descent type and show its convergence. Next we consider
minimization problems that can be transformed into \ell_pspaces. In
this case, we formulate a computable version of the adaptive algortihm
and show its convergence. We present joint work with Claudio Canuto.

CPDW03 
9th April 2003 16:30 to 17:00 
SC Hawkins 
Easily inverted approximation type preconditioners for almostdiagonal matrices arising from twodimensional elliptic operators
For elliptic operators represented in a wavelet basis there exists an optimal diagonal preconditioner that yields a condition number that is independent of basis size.
This is well known.But in practice the condition number can still be large and this must be addressed by further preconditioning.
Preconditioners based on Sparse Approximate Inverse techniques have been presented by
Cohen and Masson [Siam J. Sci. Comput., 21:10061026, 1999], and
by Chan, Tang and Wan [BIT, 37:644660, 1997] for matrices obtained by applying a discrete wavelettransform to a matrix.
Preconditioning with an easily inverted sparse approximation is considered
in the onedimensional case by Chen [ETNA, 8:138153, 1999].
We consider easily inverted sparse approximation preconditioning in the twodimensional case.
While representation in a single scale basis typically produces a matrix with diagonal structure,
representation in a wavelet basis produces a more complicated structure
sometimes called almostdiagonal, or fingerpatterned in the case of onedimensional problems.
Sparse matrices with these structures typically have dense LU factors, and so are
difficult to invert.
Chen reorders the wavelet basis for onedimensional problems to produce
what is essentially a banded matrixthe banded part of the matrix is the desired easily inverted sparse approximation because it has sparse factors that can be cheaply computed.
The ordering concentrates much of the fingerpatterned structure into the banded matrix so that it makes a good preconditioner. Preconditioners of this type have not previously been developed for twodimensional problems.

CPDW03 
9th April 2003 17:00 to 18:30 
Informal time for researchers from various disciplines to exchange ideas \& explore currently unsolved problems  
CPDW03 
10th April 2003 09:00 to 10:00 
Waveletbased adaptive optimisation
We use the properties of wavelets bases to design an adaptive descent
algorithm for solving a convex minimization problem in a function space
of Hilbert type. We prove the convergence of the algorithm and we
discuss its optimality in the framework of best Nterm approximation
results (Cohen, Dahmen and DeVore). This is a joint work with K. Urban.


CPDW03 
10th April 2003 10:00 to 11:00 
Coordinated discussion session: Multiscale methods for PDEs  
CPDW03 
10th April 2003 11:30 to 12:30 
hpfinite adaptive element methods for time harmonic Maxwell's equations
Recently, there has been considerable interest in the use of high order
finite element methods for the approximation of Maxwell's equations.
We shall survey some of our own work in this area. In particular, we
shall present families of hierarchic basis functions for the Galerkin
discretisation of the space $H({\rm curl};\Omega)$ that naturally arises
in the variational formulation of Maxwell equations. The conditioning
and dispersive behaviour of the elements is discussed along with
approximation theory. Numerical examples are shown which indicate
the potential of the methods for computing approximations of the
timeharmonic Maxwell's equations.


CPDW03 
10th April 2003 14:00 to 15:00 
C Carstensen 
Scientific computing of related minimisation models in three applications
The three applications considered for a nonconvex minimisation problem
model examples for phase transitions, optimal design tasks,
and micromagnetics. Since (quasi) convexity is essentially equivalent
to lower semicontinuity, typically, the minimiser in such
nonconvex minimisation problems is NOT attained: Infimising sequences
develop enforced oscillations of an arbitrary fine length scale and
converge weakly but not strongly to an averaged generalised
solution. The approximation
of which plus an approximation of the generated Young measures is the
task of a numerical simulation. In the direct approach to computational
microstructures, the finite element solutions develop oscillations
quite difficult to compute. In the three examples at hand,
relaxation theory provides auxiliary problems (relaxed models) which
are convex but not uniformly or even strictly convex. Their numerical
simulation, in the quasiconvexified approach to computational
microstructures, is relatively easy and recommended and even the
Young Measures generated in the nonconvex model can be recovered.
However, there is a reliabilityefficiency gap and strong convergence
in energy norms is under debate when stabilisation techniques are employed.

CPDW03 
10th April 2003 15:30 to 16:00 
Geological modelling: multiphysics data to multiscale models
The talk discusses competing approaches to modelling multiscale systems regarding flow through porous media.It will be shown that 'upscaling' is in somecircumstances a multiscale finite volume method.


CPDW03 
10th April 2003 16:00 to 16:30 
Adaptive Monte Carlo algorithm for killed diffusion  
CPDW03 
10th April 2003 16:30 to 17:00 
Convergence rates for adaptive FEM  
CPDW03 
10th April 2003 17:00 to 18:30 
Informal time for researchers from various disciplines to exchange ideas \& explore currently unsolved problems  
CPDW03 
11th April 2003 09:00 to 10:00 
J Xu 
Multilevel techniques for grid adaptation
Some recent studies will be reported in this talk on using multigrid ideas in grid adaptation. Results to be presented include gradient and Hessian recovery schemes by using averaging and smoothing (as in multigrid), interpolation error estimates for both isotropic and
anisotropic grids and multilevel techniques for global grid moving and local grid refining. Some applications will also be given. (Parts of the talk are joint works with Randy Bank, and with Long Chen and Pengtao Sun).

CPDW03 
11th April 2003 10:00 to 11:00 
Coordinated discussion session: Multigrid and multilevel methods  
CPDW03 
11th April 2003 11:30 to 12:00 
TJ Barth  Preliminary results for aposteriori error estimation and adaptivity in numerical magnetohydrodynamics 
CPDW03 
11th April 2003 12:00 to 12:30 
Anisotropic mesh refinement based upon error gradients
Many multiscale phenomena are highly anisotropic. For example planar shocks, boundary layers, solidification fronts, etc., each have particular directions for which the resolution should be finest (i.e. perpendicular to the shock/layer/front). In order to obtain an adaptive
algorithm of optimal complexity therefore the refinement must respect this local directionality. This requires two separate, but equally important, components: a practical mesh adaptivity mechanism for delivering a desired local anisotropic mesh refinement; and an error estimator that can efficiently and reliably direct this refinement.
This presentation will focus on the second of these two components by considering a strategy for driving anisotropic mesh refinement that is based upon the use the gradient of the error function, or a suitable approximation to it. Unlike with conventional error estimation, which is only designed to indicate where it is necessary to locally refine, for an estimate to be useful for anisotropic refinement it must also provide information on which direction(s) in which to refine. It will be demonstrated that the use of anisotropic adaptivity driven by the gradients of certain a posteriori error estimates is indeed feasible and
appears to offer significant potential in terms of efficiency gains that move towards an optimalcomplexity algorithm, at least for a class of linear reactiondiffusion test problems that is considered.


CPDW03 
11th April 2003 14:00 to 15:00 
B Engquist  Heterogeneous multiscale methods for dynamical systems 
CPDW03 
11th April 2003 15:00 to 15:30 
Closing discussion  
CPDW04 
14th April 2003 13:55 to 14:00 
Opening Remarks  
CPDW04 
14th April 2003 14:00 to 15:00 
A semiLagrangian scheme for mean curvature motion  
CPDW04 
14th April 2003 15:30 to 16:15 
V Styles  Numerical analysis of models for diffusion induced grain boundary motion 
CPDW04 
14th April 2003 16:15 to 17:00 
Finite element approximation of a void electromigration model  
CPDW04 
14th April 2003 17:00 to 17:45 
FA Radu  Finite element approximation of saturated/unsaturated flow and reactive solute transport in porous media 
CPDW04 
15th April 2003 09:00 to 10:00 
J Sprekels  Modelling and simulation of the sublimation growth of SiC bulk single crystals 
CPDW04 
15th April 2003 10:00 to 11:00 
R Kimmel  On active contours, edge detection and segmentation 
CPDW04 
15th April 2003 11:30 to 12:30 
Z Chen  Sharp L$^1$ a posteriori error analysis for nonlinear convectiondiffusion problems 
CPDW04 
15th April 2003 14:00 to 15:00 
G Dziuk  Numerical methods for conformally parametrized surfaces 
CPDW04 
15th April 2003 15:30 to 16:15 
CJ Heine  Computations of form and stability of rotating drops with finite elements 
CPDW04 
15th April 2003 16:15 to 17:00 
Phase field models for dendritic growth  
CPDW04 
15th April 2003 17:00 to 18:30 
Informal time for researchers from various disciplines to exchange ideas \& explore currently unsolved problems  
CPDW04 
16th April 2003 09:15 to 10:00 
U Clarenz  Surface restoration via Willmoreflow 
CPDW04 
16th April 2003 10:00 to 11:00 
Monotone multigrid for AllenCahn equations  
CPDW04 
16th April 2003 11:30 to 12:30 
Eikonal equations with discontinuities: uniqueness and numerical approximation  
CPDW04 
16th April 2003 14:00 to 15:00 
Fullcontact in a posteriori error estimation for obstacle problems  
CPDW04 
16th April 2003 15:30 to 16:15 
MB Hindmarsh  Relativistic motion by mean curvature 
CPDW04 
16th April 2003 16:15 to 17:00 
M Benes  Modified AllenCahn equation and applications 
CPDW04 
17th April 2003 10:15 to 11:00 
Two models of phase transition  
CPDW04 
17th April 2003 11:30 to 12:30 
Finite element methods for surface diffusion  
CPDW04 
17th April 2003 14:00 to 15:00 
A Schmidt  Adaptive finite element methods for phase field problems 
CPDW04 
17th April 2003 15:30 to 16:15 
O Lakkis  Computing epitaxial growth with attachmentdetachment kinetics 
CPDW04 
17th April 2003 16:15 to 17:00 
A posteriori error estimates for the AllenCahn problem: circumventing Gronwall's nightmare  
CPDW04 
17th April 2003 17:00 to 18:30 
Informal time for researchers from various disciplines to exchange ideas \& explore currently unsolved problems  
CPD 
30th April 2003 09:30 to 10:15 
Regularity and singularities for Maxwell near edges and corners of a polyhedral conductor I  
CPD 
30th April 2003 10:30 to 11:15 
Approximation of Maxwell singularities by higher order finite elements I  
CPD 
30th April 2003 11:30 to 12:15 
Eigenvalue approximation of noncoercive operators  
CPD 
30th April 2003 14:00 to 14:45 
Regularity and singularities for Maxwell near edges and corners of a polyhedral conductor II  
CPD 
30th April 2003 14:45 to 15:30 
Approximation of Maxwell singularities by higher order finite elements II  
CPD 
30th April 2003 16:00 to 16:45 
Discrete compactness and convergence of Maxwell eigenvalues  
CPDW05 
8th May 2003 11:30 to 11:50 
Finite elements on degenerate meshes  
CPDW05 
8th May 2003 11:50 to 12:10 
Image processing with partial differential equations  
CPDW05 
8th May 2003 12:10 to 12:30 
Mott MacDonald's computational hydraulic and eco system model HYDRO3D  
CPDW05 
8th May 2003 14:00 to 14:20 
Adaptive explicit timestepping for stiff ODEs  
CPDW05 
8th May 2003 14:20 to 14:40 
C Carstensen  Averaging FE error control 
CPDW05 
8th May 2003 16:30 to 16:50 
Discrete compactness property for quadrilateral finite element spaces  
CPDW05 
8th May 2003 16:50 to 17:10 
Finite element approximation of parabolic stochastic partial differential equation  
CPDW05 
8th May 2003 17:10 to 17:30 
A posteriori estimates for evolution problems and continuous reconstructions  
CPDW05 
9th May 2003 09:45 to 10:05 
Flexible finite element modelling in FEMLAB  
CPDW05 
9th May 2003 10:05 to 10:25 
B Guo  Mathematical foundation for the pversion of the finite element method 
CPDW05 
9th May 2003 10:25 to 10:45 
Optimal relaxation parameter for the Uzawa method  
CPDW05 
9th May 2003 11:30 to 11:50 
Z Chen  An adaptive FE method with perfectly matched absorbing layers for the wave scattering by periodic structures 
CPDW05 
9th May 2003 11:50 to 12:10 
Finite element approximation to evolution problems in mixed form  
CPDW05 
9th May 2003 12:10 to 12:30 
P Monk  A discontinuous Galerkin method for symmetric hyperbolic systems 
CPDW05 
9th May 2003 14:00 to 14:20 
Finite element approach to the immersed boundary method  
CPDW05 
9th May 2003 14:20 to 14:40 
C Johnson  The deep connections between Bebop and finite elements 
CPDW06 
13th May 2003 11:00 to 12:00 
M Luskin  Metastability and microstructure in structural phase transformations 
CPDW06 
13th May 2003 14:00 to 15:00 
Inserting the atomic scale in computational materials science: state of the art and challenges  
CPDW06 
13th May 2003 15:30 to 16:30 
H Garcke  Phase field models for diffusional phase transformations in multicomponent alloys 
CPDW06 
14th May 2003 10:00 to 11:00 
On the influence of mechanical fields on chemistry, diffusion and interface motion in solids  
CPDW06 
14th May 2003 11:30 to 12:30 
Approximation of lattice systems at finite temperature  
CPDW06 
14th May 2003 14:00 to 15:00 
M Katsoulakis  Coarsegrained stochastic processes and Monte Carlo simulations in lattice systems 
CPD 
4th June 2003 11:30 to 12:30 
Total variation flow  
CPD 
11th June 2003 14:30 to 15:30 
Finite element analysis of an electromagnetic problem arising from the simulation of metallurgical electrodes  
CPD 
11th June 2003 16:00 to 17:00 
RG Durań  Neumann problems for elliptic equations in domains with cusps 
CPDW07 
21st June 2003 09:00 to 18:00 
Unknown  
CPDW07 
22nd June 2003 09:00 to 18:00 
Unknown  
CPDW07 
23rd June 2003 09:00 to 18:00 
Unknown  
CPDW07 
24th June 2003 09:00 to 18:00 
Unknown  
CPD 
27th June 2003 10:00 to 11:00 
PB Bochev  Constrained interpolation algorithms for divergence free remap 