Computational challenges in partial differential equations

Programme theme

The study of partial differential equations (PDE's) is a fundamental subject area of mathematics which links important strands of pure mathematics to applied and computational mathematics. Indeed PDE's are ubiquitous in almost all of the applications of mathematics where they provide a natural mathematical description of phenomena in the physical, natural and social sciences.

Partial differential equations and their solutions exhibit rich and complex structures. Unfortunately, closed analytical expressions for their solutions can be found only in very special circumstances, and these are mostly of limited theoretical and practical interest. Thus, scientists and mathematicians have been naturally led to seeking techniques for the approximation of solutions. Indeed, the advent of digital computers has stimulated the incarnation of Computational Mathematics, much of which is concerned with the construction and the mathematical analysis of numerical algorithms for the approximate solution of partial differential equations.

The present programme focuses on some of the most exciting and promising mathematical ideas in these fields, and those branches of PDE theory that provide a source of physically relevant and mathematically hard problems to stimulate future development. The main themes of the programme are:

Cognitive algorithms: adaptivity, feed-back and a posteriori error control;
Hierarchical modelling: multi-scale mathematical models and algorithms;
Nonlinear degenerate PDEs and problems with non-smooth solutiopns and with interfaces and line singularities: fourth-order degenerate parabolic equations, fully nonlinear second-order equations, free boundary problems for unknown codimension 1 interfaces, and equations of motion for line and point defects.

The aim of this programme is to bring together experts in the numerical, applied and computational analysis of PDE's and their applications and to address several major topics of research of significant mathematical, scientific and computational importance.

Concurrent Newton Institute Programme:

Nonlinear Hyperbolic Waves in Phase Dynamics and Astrophysics

LMS-EPSRC Short Instructional Course on Numerical Analysis Durham 2002

11th Leslie Fox Prize in Numerical Analysis, Cambridge, 20th June 2003

Click here to download the programme's final scientific report

Preprints

Title	Year	Programme
An anomalous flow: everywhere locally absolutely unstable yet globally stable Authors: Robert Hunt Download PDF	2007	CPD
Poisson processes revisited Authors: John Kingman Download PDF	2006	CPD
Spectra of positive matrices and the Markov group conjecture Authors: John Kingman Download PDF	2006	CPD
Progress and problems in the theory of regenerative phenomena Authors: John Kingman Download PDF	2005	CPD
Progress and problems in the theory of regenerative phenomena Authors: John Kingman Download PDF	2005	CPD
The Poisson-Dirichlet distribution and the frequency of large prime devisors (revised) Authors: John Kingman Download PDF	2004	CPD
Stochastic Aspects of Lanchester's Theory of Warfare Authors: John Kingman Download PDF	2002	CPD
Powers and products of regenerative phenomena Authors: John Kingman Download PDF	2002	CPD
Extremal problems for regenerative phenomena Authors: John Kingman Download PDF	2002	CPD
A coercive combined field integral equation for electromagnetic scattering Authors: Ralf Hiptmair, A Buffa Download PDF	2002	CPD
An adaptive mesh-refining algorithm allowing for an ${\it H}^1$-Stable ${\it L}^2$-projection onto courant finite element spaces Authors: Carsten Carstensen Download PDF	2002	CPD
Discrete dispersion relation for ${\it hp}$-version finite element approximation at high wave number Authors: M Ainsworth Download PDF	2002	CPD
The ultra weak variational formulation for elastic wave problems Authors: P Monk, F Collino, ET Al, T Huttunen Download PDF	2002	CPD
Robust a posteriori error estimation for non-conforming finite element approximation Authors: ET Al, M Ainsworth Download PDF	2002	CPD
Explicit and averaging a posteriori error estimates for adaptive finite volume methods Authors: R Lazarov, S Tomov, Carsten Carstensen Download PDF	2002	CPD
Numerical solution of parabolic equations in high dimensions Authors: C Schwab, T Von Petersdorff Download PDF	2002	CPD
Stability and geometric conservation laws for ALE formulations Authors: D Boffi, Lucia Gastaldi Download PDF	2002	CPD
On the choice of a stabilizing subgrid for convection-diffusion problems Authors: Franco Brezzi, Donatella Marini, A Russo Download PDF	2002	CPD
Singularities of eddy current problems Authors: S Nicaise, M Costabel, Monique Dauge Download PDF	2002	CPD
Stability and convergence of collocation schemes for retarded potential integral equations Authors: Dougald Duncan, Penny Davies Download PDF	2002	CPD
On explicit time-stepping for stiff odes Authors: Claes Johnson, Anders Logg, K Eriksson Download PDF	2002	CPD
Computational solution, applications and analysis of some geometrical nonlinear diffusion equations Authors: Karol Mikula Download PDF	2002	CPD
On linearized backward Euler method for the equations of motion arising in the Oldroyd fluids of order one. Authors: PD Damazio, JY Yuan, Amiya Kumar Pani Download PDF	2002	CPD
An eddy current and micromagnetism model with applications to disk write heads Authors: F Collino, Kui-Hua Sun, PB Monk, ET Al Download PDF	2002	CPD
The optimal convergence of the ${\it h-p}$ version of the boundary element method with quasiuniform meshes for elliptic problems on polygonal domains Authors: Benqui Guo, N Heuer Download PDF	2002	CPD
Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities Authors: Charlie Elliott, K Deckelnick Download PDF	2002	CPD
Best approximation for the p-version of the finite element method in three dimensions in the framework of the Jacobi-weighted Besov spaces Authors: Benqui Guo Download PDF	2002	CPD
Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model Authors: JY Yuan, Amiya Kumar Pani Download PDF	2002	CPD
A posteriori error analysis for the mean curvature flow of graphs Authors: Ricardo Nochetto, O Lakkis Download PDF	2002	CPD
Finite element analysis of a current density - electric field formulation of Bean's model for superconductivity Authors: David Kay, Vanessa Styles, Charlie Elliott Download PDF	2002	CPD
Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods Authors: M Ainsworth Download PDF	2002	CPD
Discrete dynamics for convex and non-convex smoothing functionals in PDE based image restoration Authors: S Maier-Paape, Charlie Elliott, B Gawron, ET Al Download PDF	2002	CPD
An efficient solution to the Eikonal equation on parametric manifolds Authors: R Kimmel, A Spira Download PDF	2002	CPD
The computation of conical diffraction coefficients in high-frequency acoustic wave scattering Authors: Ivan Graham, BD Bonner, VP Smyshlyaev Download PDF	2002	CPD
Finite element methods in local active control of sound Authors: P Gamallo, Rodolfo Rodriguez, A Bermudez Download PDF	2002	CPD
A wavenumber independent boundary element method for an acoustic scattering problem Authors: SN Chandler-wilde, S Langdon Download PDF	2002	CPD
${\it HP}$-interpolation of non-smooth functions Authors: Markus Melenk Download PDF	2002	CPD
Finite element approximation of a Stefan problem with degenerate Joule heating Authors: R Nurnberg, John Barrett Download PDF	2002	CPD
AC losses in type-II superconductors induced by non-uniform fluctuations of external magnetic field Authors: V Sokolovsky, Leonid Prigozhin Download PDF	2002	CPD
A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media Authors: GR Richter, P Monk Download PDF	2002	CPD
Sharp $\it L^1$ a posteriori error analysis for nonlinear convection diffusion problems Authors: G Ji, Zhiming Chen Download PDF	2002	CPD
Fully discrete semi-implicit second order splitting for anisotropic surface diffusion of graphs Authors: Gerhard Dziuk, Charlie Elliott, K Deckelnick Download PDF	2002	CPD
Finite element approximation of a nonlinear cross-diffusion population model Authors: James Blowey, John Barrett Download PDF	2002	CPD
Reliable a posteriori error control for non-conforming finite element approximation of Stokes flow Authors: M Ainsworth, Willy Dörfler Download PDF	2002	CPD
Poincaré-type inequalities for broken Sobolev spaces Authors: Endre Süli, Andris Lasis Download PDF	2002	CPD
${\it HP}$-version discontinuous Galerkin finite element methods for semlinear parabolic problems Authors: Endre Süli, Andris Lasis Download PDF	2002	CPD
Three Matlab implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control Authors: Carsten Carstensen, C Bahriawati Download PDF	2002	CPD
Effective stimulation of a macroscopic model for stationary micromagnetics Authors: Dirk Praetorius, Carsten Carstensen Download PDF	2002	CPD
Effective relaxation for microstructure simulations: algorithms and applications Authors: Carsten Carstensen, Klaus Hackl, ET Al, Sören Bartels Download PDF	2002	CPD
Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid Authors: H Garcke, R Nurnberg, John Barrett Download PDF	2002	CPD
Approximation theory for the ${\it p}$-version of the finite element method in three dimensions in the framework of the Jacobi-weighted Besov spaces Authors: Benqui Guo Download PDF	2002	CPD
Existence of global weak solutions for some polymeric flow models Authors: C Schwab, Endre Süli, John Barrett Download PDF	2002	CPD
Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation Authors: Xiaobing Feng, Andreas Prohl, John Barrett Download PDF	2002	CPD

Workshops

CPDW01

Seminars

Wednesday 5th February 2003
15:30 to 16:30	David Silvester Fast black-box preconditioners for elliptic PDE problems	Room 1
17:00 to 18:00	William McLean University of New South Wales Time discretisation of an evolution equation via Laplace transforms	Room 1

Thursday 6th February 2003
15:00 to 16:00	Sparse finite elements for PDEs with stochastic data	Room 1

Wednesday 19th February 2003
15:30 to 16:30	Hiptmair ETH Zürich FEM-BEM coupling for quasistatic electromagnetism	Room 1

Wednesday 12th March 2003
16:00 to 17:00	An adaptive finite element method for minimal surfaces	Room 1

Wednesday 19th March 2003
16:00 to 17:00	Michael Hinze A generalised discretisation concept for optimal control with pdes	Room 1

Wednesday 26th March 2003
16:00 to 17:00	Andreas Veeser University of Milan Convergent adaptive finite elements for the p-laplacian	Room 1

Wednesday 30th April 2003
09:30 to 10:15	Dauge Université de Rennes 1 Regularity and singularities for Maxwell near edges and corners of a polyhedral conductor I	Room 2
10:30 to 11:15	Costabel Université de Rennes 1 Approximation of Maxwell singularities by higher order finite elements I	Room 2
11:30 to 12:15	Daniele Boffi Eigenvalue approximation of noncoercive operators	Room 2
14:00 to 14:45	Dauge Université de Rennes 1 Regularity and singularities for Maxwell near edges and corners of a polyhedral conductor II	Room 2
14:45 to 15:30	Costabel Université de Rennes 1 Approximation of Maxwell singularities by higher order finite elements II	Room 2
16:00 to 16:45	Daniele Boffi Discrete compactness and convergence of Maxwell eigenvalues	Room 2

Wednesday 14th May 2003
10:00 to 11:00	Wolfgang Dreyer On the influence of mechanical fields on chemistry, diffusion and interface motion in solids	Room 1
11:30 to 12:30	Petr Plechac University of Warwick Approximation of lattice systems at finite temperature	Room 1
14:00 to 15:00	Coarse-grained stochastic processes and Monte Carlo simulations in lattice systems	Room 1

Wednesday 4th June 2003
11:30 to 12:30	Xiaobing Feng University of Tennessee Total variation flow	Room 1

Wednesday 11th June 2003
14:30 to 15:30	Finite element analysis of an electromagnetic problem arising from the simulation of metallurgical electrodes	Room 1
16:00 to 17:00	Neumann problems for elliptic equations in domains with cusps	Room 1

Friday 27th June 2003
10:00 to 11:00	Pavel Bochev Sandia National Laboratories Constrained interpolation algorithms for divergence free remap	Room 2

Computational challenges in partial differential equations

Organisers

Mark Ainsworth
Charlie Elliott University of Sussex
Endre Süli University of Oxford

Speakers

Pavel Bochev Sandia National Laboratories
Daniele Boffi
Costabel Université de Rennes 1
Dauge Université de Rennes 1
Wolfgang Dreyer
Xiaobing Feng University of Tennessee
Michael Hinze
Hiptmair ETH Zürich
William McLean University of New South Wales
Petr Plechac University of Warwick
David Silvester
Andreas Veeser University of Milan

Show All

Computational challenges in partial differential equations

Programme theme

Preprints

An anomalous flow: everywhere locally absolutely unstable yet globally stable

Poisson processes revisited

Spectra of positive matrices and the Markov group conjecture

Progress and problems in the theory of regenerative phenomena

Progress and problems in the theory of regenerative phenomena

The Poisson-Dirichlet distribution and the frequency of large prime devisors (revised)

Stochastic Aspects of Lanchester's Theory of Warfare

Powers and products of regenerative phenomena

Extremal problems for regenerative phenomena

A coercive combined field integral equation for electromagnetic scattering

An adaptive mesh-refining algorithm allowing for an ${\it H}^1$-Stable ${\it L}^2$-projection onto courant finite element spaces

Discrete dispersion relation for ${\it hp}$-version finite element approximation at high wave number

The ultra weak variational formulation for elastic wave problems

Robust a posteriori error estimation for non-conforming finite element approximation

Explicit and averaging a posteriori error estimates for adaptive finite volume methods

Numerical solution of parabolic equations in high dimensions

Stability and geometric conservation laws for ALE formulations

On the choice of a stabilizing subgrid for convection-diffusion problems

Singularities of eddy current problems

Stability and convergence of collocation schemes for retarded potential integral equations

On explicit time-stepping for stiff odes

Computational solution, applications and analysis of some geometrical nonlinear diffusion equations

On linearized backward Euler method for the equations of motion arising in the Oldroyd fluids of order one.

An eddy current and micromagnetism model with applications to disk write heads

The optimal convergence of the ${\it h-p}$ version of the boundary element method with quasiuniform meshes for elliptic problems on polygonal domains

Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities

Best approximation for the p-version of the finite element method in three dimensions in the framework of the Jacobi-weighted Besov spaces

Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model

A posteriori error analysis for the mean curvature flow of graphs

Finite element analysis of a current density - electric field formulation of Bean's model for superconductivity

Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods

Discrete dynamics for convex and non-convex smoothing functionals in PDE based image restoration

An efficient solution to the Eikonal equation on parametric manifolds

The computation of conical diffraction coefficients in high-frequency acoustic wave scattering

Finite element methods in local active control of sound

A wavenumber independent boundary element method for an acoustic scattering problem

${\it HP}$-interpolation of non-smooth functions

Finite element approximation of a Stefan problem with degenerate Joule heating

AC losses in type-II superconductors induced by non-uniform fluctuations of external magnetic field

A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media

Sharp $\it L^1$ a posteriori error analysis for nonlinear convection diffusion problems

Fully discrete semi-implicit second order splitting for anisotropic surface diffusion of graphs

Finite element approximation of a nonlinear cross-diffusion population model

Reliable a posteriori error control for non-conforming finite element approximation of Stokes flow

Poincaré-type inequalities for broken Sobolev spaces

${\it HP}$-version discontinuous Galerkin finite element methods for semlinear parabolic problems

Three Matlab implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control

Effective stimulation of a macroscopic model for stationary micromagnetics

Effective relaxation for microstructure simulations: algorithms and applications

Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid

Approximation theory for the ${\it p}$-version of the finite element method in three dimensions in the framework of the Jacobi-weighted Besov spaces

Existence of global weak solutions for some polymeric flow models

Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation

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