MATH 689: Preconditioning techniques in finite element methods, Spring 2002

General Information:

Instructor: James H. Bramble,

Office Hours: By appointment

Blocker 505, Phone: 845-7137,

bramble@math.tamu.edu

Time/Classroom:: Wed. 8:45-11:15 Blocker 624

Notice:

Students with disabilities can get assistance from the Office of Services for Students with Disabilities (845-1637)

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Preconditioning is a critical aspect for the development of efficient computational algorithms for large scale scientific computations. This course will cover a broad range of techniques for the construction and analysis of preconditioners for the discrete systems which arise from finite element approximations of boundary value problems.

Much of the material will come from research papers although some can be found in the following:

Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations by B. Smith, P. Bjorstad, and W. Gropp, Cambridge University Press, 1996.
Domain Decompositions Methods for Partial Differential Equations by A. Quarteroni and A. Valli, Oxford Science Publications, 1999.
The Analysis of Multigrid Methods by James H. Bramble and Xuejun Zhang. In Handbook of Numerical Analysis VII, Elsevier, 2000.

Course Outline (subject to adjustments):

Preliminary material including model problems and Sobolev spaces; interpolation spaces and fractional order spaces.
Multilevel methods. Representation of norms.
Multilevel preconditioners.
Schwarz methods; the theory of subspace iterations.
Non-overlapping domain decomposition preconditioners in two space dimensions.
Non-overlapping domain decomposition preconditioners in three spatial dimensions.
Two level methods for nonconforming applications.

Grading Policy:

The course grade will be determined from homework and class presentation. Each student will be required to read and present the results from a research paper.

Prerequisites:

Math 609-610 or equivalent.