l. Short Course title:

Adaptive Methods for PDE-solution

2. Date:

February 9, 2003
SIAM Associated Conference: CSE03

3. Short Course Organizers:

David Brown
Center for Applied Scientific Computing
Lawrence Livermore National Laboratory
Livermore, CA 94551
925-424-3557 (phone)
925-424-2477 (fax)

Lori Freitag
Mathematics and Computer Science Division
Argonne National Laboratory
Argonne, IL 60439
630-252-7245 (phone)
630-252-5986 (fax)

4. Rationale

The use of adaptive methods is commonly recognized as a critical aspect in the numerical solution of large-scale PDE-based. To meet this need, much work has been done to develop algorithms and software for various adaptive strategies, including spatial (or h-) refinement, adaptive polynomial order (p-refinement), and interface tracking.

Within each of these broad categories there are further classifications; for example h-refinement can take many different forms ranging from block-structured, patch-based methods to unstructured techniques that are both conformal and nonconformal. Given the wide array of choices available to application scientists, it is often unclear which technique is best suited for a given problem. This course is designed to introduce three commonly-used methods and compare their advantages
and disadvantages for different classes of problems.

5. Lecturers

Because this course is intended to be a survey of different adaptive techniques, we have invited three lecturers with expertise in different major categories of adaptivity. In particular, Phil Colella is an expert in block structured and embedded boundary techniques, Mark Shephard is an expert in unstructured h- and h-p adaptivity, and Jim Glimm is an expert in front tracking procedures. Each lecturer has been asked to give an overview of the widely used techniques and software in their area and to provide information and pointers to advanced material where attendess can learn more details.

Phillip Colella
Lawrence Berkeley National Laboratory

Bio: Phil Colella is the group leader of the Applied Numerical Algorithms Group at the National Energy Research Scientific Computing Center at Lawrence Berkeley National Lab. He is widely recognized for his contributions in high-resolution finite difference methods, adaptive mesh refinement, volume-of-fluid methods for fronts and irregular geometries, and multidimensional shock dynamics.

James Glimm
SUNY Stony Brook, Brookhaven National Laboratory

Bio: James Glimm received his Ph D from Columbia University. He is widely known for work in a number of fields of pure and applied mathematics and physics. His contributions to numerical analysis have been centered on methods for front tracking. He is one of the developers of the Front Tracking software package FronTier.

Mark Shephard
Rensselaer Polytechnic Institute

Bio: Mark Shephard is the Samuel A. and Elisabeth C. Johnson, Jr. Professor of Engineering and Director of the Scientific Computation Research Center at Rensselaer Polytechnic Institute. He has published over 225 papers in the area of automated and adaptive finite element modeling, and is editor of Engineering with Computers and on the editorial board of five computational mechanics journals.

6. Description

We have invited three world-class experts to introduce commonly-used techniques for adaptive solution of PDE-based applications, namely h-p unstructured adaptive mesh refinement, block structured adaptive mesh refinement, and interface tracking. Each method will be discussed in detail in two-hour, focused segments during which the basic techniques will be described, the advantages and disadvantages of each method will be elucidated, and a comparison with the other techniques will be given. During each lecture, several examples will highlight the strengths of the method and pointers to available software will be given. At the end of the individual lectures, the authors will participate in a panel discussion.

7. Course Objectives

This course is intended to provide an overview of three different strategies for adaptive mesh techniques: unstructured h-p adaptivity, block structured adaptivity, and front tracking techniques. Application scientists interested in exploring these methods for their problems will be introduced to the advantages and disadvantages
of each technique as well as the publicly available software in each area.

8. Level of Material

Introductory 50%
Intermediate 35%
Advanced 15%

9. Who Should Attend?

The intended audience is computational scientists who are interested in mesh-based, adaptive methods in the solution of PDE applications.

10. Recommended Background

Some basic knowledge of numerical solution of PDEs is required; attendees should be familiar with mesh-based discretization schemes.

11. Course Outline

1. Block Structured Adaptivity

2. H- and H-P Adaptivity for Unstructured Meshes

3. Front Tracking Techniques

Pointers to further information

Front Tracking:
1. J. Glimm, J. W. Grove, X.-L. Li, K.-M. Shyue, Q. Zhang and Y. Zeng, "Three Dimensional Front Tracking". SIAM J. Sci. Comp. 19-3 (1998), 703--727.

2. J. Glimm, J. W. Grove, X.-L. Li and D. C. Tan, "Robust Computational Algorithms for Dynamic Interface Tracking in Three Dimensions". SIAM J. Sci. Comp. 21-1 (2000), 2240--2256.

3. J. Glimm, J. W. Grove and Y. Zhang, "Interface Tracking for Axisymmetric Flows". SIAM J. SciComp. 24-1 (2002), 208--236.