Workshop at Newton Institute Heralds Programme on Computational Challenges in PDEs

May 31, 2003


With Newton looking on, the organisers and international advisers gather at the conclusion of the successful opening workshop at the Isaac Newton Institute for Mathematical Sciences. Left to right: Ricardo Nochetto, Mark Ainsworth, Endre Süli, Charlie Elliott, and John Barrett.

Mark Ainsworth, Charlie Elliott, and Endre Süli

The Isaac Newton Institute for Mathematical Sciences (based at the University of Cambridge, UK) marked the start of a six-month initiative on the computational challenges of partial differential equations with a workshop, "Mathematical Challenges in Science and Engineering Computation," January 20-24. This is the first programme devoted to numerical analysis and computation at the Newton Institute, which recently celebrated its tenth anniversary. The workshop drew more than 80 participants, from the UK and overseas, and speakers discussed each of the three main topics of the full six-month programme: error control and adaptive algorithms, hierarchical and multiscale modelling, and interfaces and free boundaries.

Rolf Rannacher (Universität Heidelberg) opened the workshop with a discussion of error control and adaptivity in numerical computation, and described applications to numerical approximation of chemically reacting flows. Taking up these issues in the context of free boundary problems, Ricardo Nochetto (University of Maryland) observed that many of the difficulties revolve around identification of an appropriate notion of residual for such problems; he showed how error control is accomplished for a range of examples, including nonlinear degenerate parabolic equations, mean-curvature flow problems, and surface diffusion. The day concluded with an open discussion of the role of computational mathematics in complex industrial simulations, chaired by John Ockendon of Oxford University, followed by a reception at which overseas participants were invited to sample traditional "warm" local ale!

Introducing the theme of the second day-problems involving interfaces-Olivier Pironneau (Université Paris VI) described his recent work on the sensitivity of shocks that are present in solutions of Euler equations and are important in computations of flutter. Charlie Elliott of the University of Sussex presented a variety of physical problems in which curvature-dependent evolution of interfaces plays a lead role, including diffusion-induced grain boundary motion, along with an overview of the principal mathematical models. Adam Wheeler (Southampton University) discussed diffuse-interface, or phase-field, models and their application to solidification; Adrian Sutton (Oxford University) made a case for the use of simulation at the atomistic level.

Atomistic and multiscale modelling of materials were the topic of the following day. In a discussion of the widely varying length scales encountered in materials science, Ellad Tadmor (Technion) described the "quasi-continuum" approach whereby adaptive techniques are used to identify different local levels of resolution. Björn Engquist (Princeton) continued the theme with a talk on his joint work with Weinan E (also of Princeton) on the heterogeneous multiscale method; E later described some of the new mathematics underlying the method. Highlighting the importance of constitutive laws at the molecular level, Tom McLeish (Leeds University) gave examples from polymer processing in which differences in structure at the molecular level are responsible for completely different macroscopic behaviours.

The penultimate day saw an influx of new participants---UK researchers involved in a five-year initiative on computational engineering mathematics supported by the Engineering and Physical Sciences Research Council of Great Britain (EPSRC), which also generously provided additional funding for the workshop. The day's programme was devoted to projects sponsored under this initiative, following opening talks by Roland Glowinski (Houston) on numerical methods for eikonal systems, and Tony Chan (UCLA) on wavelet and PDE techniques for image compression.

In a change of pace on the final day, Alfio Quarteroni (EPFL, Lausanne) discussed applications (and benefits) of numerical modelling of the cardiovascular system, a problem that requires coupled, heterogeneous PDE models of the entire system. Peter Monk (Delaware) then gave an overview of numerical techniques for the solution of inverse problems and described a new technique, the linear sampling method, that he has developed with David Colton. Closing the meeting, Russel Davies (Aberystwyth) discussed PDEs in the food industry and Nigel Weatherill (Swansea) considered issues of mesh generation in computational modelling.

The organisers of the six-month programme at the Newton Institute are Mark Ainsworth (Strathclyde), Charlie Elliott (Sussex), and Endre Süli (Oxford), assisted by an international advisory committee (John Barrett, Franco Brezzi, Ricardo Nochetto, Rolf Rannacher). Subsequent workshops in the programme will develop the topics highlighted at the opening meeting, along with new themes. The programme will also include the first European Finite Element Fair, inspired by the Finite Element Circus in the U.S. Full details can be found at www.newton.cam.ac.uk/programs/CPD/index.html.

Mark Ainsworth, Charlie Elliott, and Endre Süli are professors of mathematics at the University of Strathclyde in Glasgow, the University of Sussex, and Oxford University, respectively.


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