Ivo Babuska
Title: Computational Science, Mathematics and Where Are We Going
Abstract: Today computational science is essential tool for predicting various phenomena of interest in many fields such as engineering, physics, biology, medicine etc. The question is: Are these predictions reliable? ( Consider for example prediction of global warming, safety of atomic waste disposal etc.) Is the mathematics and numerics the major bottleneck? What is the major bottleneck? How does education reacts to this situation and needs?
The presentation will focus on these and related questions. It will address
the field of Verification and Validation and its mathematical aspects. Illustrative engineering examples will be presented.
In the classical approaches it is assumed that all needed information are
perfectly known. Unfortunately this is not realistic. Practically
all necessary information is never perfectly known. What do we do? We
will discuss this on a particular simple engineering example and associated
mathematical problem and methodology to solve it.
Walter Strauss
Title: Time-Periodic Scatterers
Abstract: Consider the wave equation in 3D perturbed by a time-periodic
potential or obstacle. I will present a survey of this subject which owes
so much to the work of Jeffery Cooper. The asymptotic behavior of the
solutions is governed by the scattering frequencies. In particular, they
are related to the question of whether or not the amplitudes of the waves
can grow unboundedly.
Douglas Arnold
Title: Stable discretizations of partial differential equations and their geometrical foundations
Abstract: Partial differential equations (PDE) are among the most useful mathematical modeling tools, and numerical discretization of PDE--approximating them by problems which can be solved on computers--is one of the most important and widely used approaches to simulating the physical world. A vastly developed technology is built on such discretizations. Nonetheless, fundamental challenges remain in the design and understanding of effective methods of discretization for certain important classes of PDE problems.
The accuracy of a simulation depends on the consistency and stability of the discretization method used. While consistency is usually elementary to establish, stability of numerical methods can be subtle, and for some key PDE problems the development of stable methods is extremely challenging. After illustrating the situation through simple (but surprising) examples, we will describe a powerful new approach--the finite element exterior calculus--to the design and understanding of discretizations for a variety of elliptic PDE problems. This approach achieves stability by developing discretizations which are compatible with the geometrical and topological structures, such as de Rham cohomology and Hodge decompositions, which underlie well-posedness of the PDE problem being solved.