Discontinuous
Galerkin Methods
Dr.
Bernardo Cockburn, University of Minnesota
Dr. Joseph E. Flaherty, Rensselaer Polytechnic Institute
To
navigate, use pull-down menus above or go to: 7th
USNCCM Site Map
Pre-Congress
Short Course
Sunday, July 27, 2003
1:00 am - 6:00 pm (5 hours)
Albuquerque Hyatt Hotel
(Conference Hotel)
Course
Description
Abstract. The
discontinuous Galerkin (DG) method provides an appealing approach
to address problems having discontinuities, such as those that arise
in hyperbolic conservation laws. Originally developed for neutron
transport problems and first analyzed by Le Saint and Raviart, the
technique lay dormant for approximately fifteen years before becoming
popular. It is now being used to solve ordinary differential equations
and hyperbolic, parabolic, and elliptic partial differential equations.
The method may
be regarded as cross between a finite volume and finite element method
and it has many of the good properties of both. Thus, for example
(i) it can sharply capture solution discontinuities relative to a
computational mesh; (ii) it simplifies adaptation since inter-element
continuity is neither required for h-refinement (mesh refinement and
coarsening) nor p-refinement (method order variation); (iii) it conserves
the appropriate physical quantities (e.g., mass, momentum, and energy)
on an elemental basis; (iv) it can handle problems in complex geometries
to high order; (v) regardless of order, it has a simple communication
pattern to elements sharing a common face that simplifies parallel
computation. With a discontinuous basis, however, the DG method produces
more unknowns for a given order of accuracy than traditional finite
element or finite volume methods and this may lead to some inefficiency.
Limiting strategies to reduce spurious oscillations when high-order
methods are applied to problems with discontinuities is difficult
when DG methods are applied on unstructured meshes.
We describe
several theoretical and computational aspects of the DG method as
it applies to hyperbolic, parabolic, and elliptic problems. Our focus
is on fluid dynamics; however, the method is capable of handling virtually
any other application. We develop the method using a local formulation
of Cockburn and Shu and describe choices for bases, numerical flux
functions, solution limiting and stabilization, shock and boundary
layer detection, local time stepping, and a posteriori error estimation.
Discontinuity detection reduces the need for limiting; thereby, retaining
a high order of accuracy in regions where solutions are smooth. We
further describe environments and data structures for serial and parallel
adaptive computation, adaptive h- and p-refinement procedures on structured
and unstructured meshes, and anisotropic mesh refinement. These techniques
are illustrated for transient compressible flow problems.
Registration
Fee $200
The registration
fee includes the following:
Instructors
Bernardo Cockburn is a
Professor of Mathematics at the University of Minnesota. He received
a 3rd-cycle Doctorate from Paris VI/XI/INRIA in Applied Mathematics
in 1983 and a PhD in Mathematics from the University of Chicago in
1986. After a one-year post-doctorate in the Institute of Mathematics
and its Applications, he joined the faculty of the School of Mathematics
at the University of Minnesota in 1987.
Professor Cockburn's research
interests focus on the development and theoretical analysis of efficient
methods for numerically solving linear and non-linear partial differential
equations. In particular, since 1987, he has been a pioneer in the
development and analysis of discontinuous Galerkin mehtods for purely
hyperbolic equations, for both compressible and incompressible flow
for high and low Reynolds numbers, and more recently for second-order
and fourth-order elliptic problems arising in mechanical and civil
engineering.
Professor
Flaherty conducts research in computational science with an emphasis
on the automatic solution of large-scale problems involving partial
differential equations. He is a pioneer in the development and use of
adaptive mesh-refinement, order variation, and mesh moving techniques
for the solution of these problems. He also has extensive contributions
in error estimation methodology and parallel computation.He
has published more than 160 papers on these and related subjects. His
work has been regularly supported by government and private organizations.
He is a fellow and Vice President of the U.S. Association for Computational
Mechanics, a member of the Executive Board of the Institute of Mathematics
and Computer Science, and a member of SIAM, ACM, and IEEE. He is editor
of Applied Numerical Mathematics, SIAM Review (Problems and Techniques
Section), and the SIAM Monographs on Mathematical Modeling and Computation.
He is on the Editorial Boards of Computational Mechanics Advances, the
International Journal of Computational Engineering and Science, and
the SIAM Journal on Scientific Computing. He received a Director's Award
from IBM, a 15 year Service Recognition from the U.S. Army.
Dr. Joseph
E. Flaherty
Scientific Computation Research Center
Rensselaer Polytechnic Institute
Troy, NY 12180 USA
Email: flahej@rpi.edu
|
