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Seminars (CPD)

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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:
  • flow through porous media with several geological layers,
  • transonic and supersonic flow with shocks,
  • accoustics with sonic boom,

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 Phase-field 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 long-range 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 long-range 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.

    (1) Direct parametrisation.
    (2) Level set method.
    (3) Phase field method - a degenerate Cahn-Hilliard equation.

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
Multiple-scale 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 atomic-scale 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 atomic-scale 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, cross-slip and jog-drag of screw dislocations in copper, stress-induced phase transformations in silicon due to nanoindentation, polarization switching in ferroelectric lead-titanate 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 macro-scale and the missing data is supplied by micro-scale 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 Non-Newtonian flow of polymer melts and solutions in complex geometries generates very novel fluid mechanics: non-inertial turbulence, elastic vortex growth, elastic free-surface 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 fixed-grid 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 multi-scale method (HMM) is a general methodology for the efficient numerical computation of problems with multi-scales and multi-physics. 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 quasi-continuum 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 residual-based 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 hp-adaptive algorithms for first-order hyperbolic systems and second-order partial differential equations of mixed elliptic-hyperbolic-parabolic 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 Non-Oscillatory (ENO) schemes for numerical shock capturing. ENO-wavelet 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 ENO-wavelet transform and obtained a rigorous approximation error bound which shows that the error in the ENO-wavelet 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 droplet-gas 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 fluid-structure 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 ill-posed 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 non-linear 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
Liquid-like 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 Navier-Stokes equations. For example, creamy soups, ketchup and yoghurt are shear-thinning; concentrated starches are shear-thickeining; 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 non-Newtonian 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 state-of-the-art 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 black-box 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 3-dimensional 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 ultra-fast 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 sub-nanosecond 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 well-accepted model of Landau and Lifshitz. Over the last years, different strategies have been developed to tackle the related non-convex 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 Landau-Lifschitz equations: existence and asymptotic behaviour
In this talk, we describe existence results for weak and regular solutions of Landau-Lifschitz 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 current-micromagnetic 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 Ginzburg-Landau to Gross-Pitaevskii
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 Bose-Einstein condensates.

In this talk, we will discuss some questions related to quantized vortices based on the Ginzburg-Landau models for superconductivity and the Gross-Pitaevskii equations for the Bose-Einstein 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 critical-state problems in superconductivity
CPD 19th February 2003
15:30 to 16:30
FEM-BEM 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 inertia-dominated 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 p-laplacian
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 advection-diffusion or advection-reaction-diffusion 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 multi-physics 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 semi-analytic 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 two-phase flow and incompressible Euler and Navier-Stokes equations with multiscale solutions. In a related effort, we introduce a new class of numerical methods to solve the stochastically forced Navier-Stokes equations. We will demonstrate that our numerical method can be used to compute accurately high order statitstical quantites more efficiently than the traditional Monte-Carlo 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 inf-sup condition; unstable elements are thus nonlinearly stabilized. We also deal with an augmented Lagrangian formulation for the Raviart-Thomas 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 quasi-optimal. 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 quasi-optimal 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 saddle-point boundary value problem which is discretized by a Galerkin finite element method. The accuracy of this discretization is controlled by residual-based 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
Co-ordinated discussion session: Adaptivity
CPDW03 8th April 2003
11:30 to 12:30
C Johnson Adaptive computational modelling of reaction-diffusion in laminar and turbulent incompressible flow
We present a framework for adaptive computational modeling with application to reaction-diffusion 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
hp-adaptive discontinuous Galerkin methods with interior penalty for degenerate elliptic partial differential equations
We consider the a posteriori and a priori error analysis of the hp-version of the discontinuous Galerkin finite element method with interior penalty for approximating second-order 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 user-defined tolerance. A key question in hp-adaptive algorithms is how to automatically decide when to h-refine/derefine and when to p-refine/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 hp-refinement algorithm is demonstrated through a series of numerical experiments. In particular, we demonstrate the superiority of using hp-adaptive mesh refinement with the traditional h-refinement 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 phase-field 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 non-conforming 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 right-hand 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
Co-ordinated 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 wavelet-best N-term 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 infinite-dimensional (exact) algorithm of steepest descent type and show its convergence. Next we consider minimization problems that can be transformed into \ell_p-spaces. 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 almost-diagonal matrices arising from two-dimensional 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., 21:1006--1026, 1999], and by Chan, Tang and Wan [BIT, 37:644--660, 1997] for matrices obtained by applying a discrete wavelettransform to a matrix. Preconditioning with an easily inverted sparse approximation is considered in the one-dimensional case by Chen [ETNA, 8:138--153, 1999]. We consider easily inverted sparse approximation preconditioning in the two-dimensional 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 almost-diagonal, or finger-patterned in the case of one-dimensional problems. Sparse matrices with these structures typically have dense LU factors, and so are difficult to invert. Chen reorders the wavelet basis for one-dimensional problems to produce what is essentially a banded matrix---the 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 finger-patterned structure into the banded matrix so that it makes a good preconditioner. Preconditioners of this type have not previously been developed for two-dimensional 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
Wavelet-based 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 N-term approximation results (Cohen, Dahmen and DeVore). This is a joint work with K. Urban.
CPDW03 10th April 2003
10:00 to 11:00
Co-ordinated discussion session: Multiscale methods for PDEs
CPDW03 10th April 2003
11:30 to 12:30
hp-finite 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 time-harmonic 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 reliability-efficiency 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
Co-ordinated discussion session: Multigrid and multilevel methods
CPDW03 11th April 2003
11:30 to 12:00
TJ Barth Preliminary results for a-posteriori 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 optimal-complexity algorithm, at least for a class of linear reaction-diffusion 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 semi-Lagrangian 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 convection-diffusion 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
C-J 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 Willmore-flow
CPDW04 16th April 2003
10:00 to 11:00
Monotone multigrid for Allen-Cahn 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
Full-contact 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 Allen-Cahn 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 attachment-detachment kinetics
CPDW04 17th April 2003
16:15 to 17:00
A posteriori error estimates for the Allen-Cahn 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 HYDRO-3D
CPDW05 8th May 2003
14:00 to 14:20
Adaptive explicit time-stepping 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 p-version 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 multi-component 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 Coarse-grained 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
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons