Below are the abstracts for the main talks at the 2005 SC Days. Links to pdf files of presentations will appear, as they become available.
Jan S Hesthaven,
Brown University
Discontinuous Galerkin Methods for Solving Time-Dependent
PDEs: Theoretical Essentials and Practical Aspects
We will discuss the basic theoretical properties and practical aspects of using discontinuous Galerkin methods for solving PDEs. These methods, although proposed first more than 3 decades ago, have recently received considerable attention due to a number of very attractive properties, e.g., solid theoretical foundation, ability to work with high-order and adaptive grids, support for unstructured grids and very high performance on parallel computers.
We shall briefly discuss some key theoretical results in the first lecture while in the second lecture we focus on more applied aspects and how to develop and implement such methods for a variety of problems and applications.
Jay Gopalakrishnan,
University of Florida
Multigrid methods and applications to electromagnetics
Multigrid methods are a class of numerical techniques to solve linear systems arising from discretization of PDEs using a heirarchy of discretization grids. These methods can often compute an approximate solution up to a given precision at asymptotically optimal computational cost. The optimal complexity of multigrid has brought within the reach of simulation many scientific problems previously thought to be of intractable size.
In the first part, we will see the mechanics of multigrid methods and explain why it leads to optimal algorithms, illustrating it on simple examples. In the second part, we will consider more complicated applications and investigate the modifications needed to successfully apply the multigrid paradigm. We will emphasize the interplay between convergence theory of discretizations and convergence analysis of multigrid and specifically how one has lead to improvements in the other. Examples from electromagnetics will be highlighted.