MATH 610-600: Numerical Methods for PDE's, Spring 2010

General Information:

• Instructor: Dr. Vivette Girault, Blocker 505B, 845-7137, girault@math.tamu.edu

• Time/Classroom:

  MWF-11:30am-12:20pm : Zachary 119D and

  TR- 9:35-10:25am : Blocker 123 (computer lab by TA)

• Office Hours: MF 10:00-11:00am or by appointment
• TA: TBA
• Textbook:Numerical Treatment of Partial Differential Equations by C. Grossmann, H. Roos and M. Stynes, (Universitext) Springer. ISBN 978-3-540-71582-5

References:

• Numerical Methods for Elliptic and Parabolic Partial Differential Equations, by Peter Knabner and Lutz Angerman (Texts in Applied Mathematics) Springer. ISBN 0-387-95449-X
• Theory and Practice of Finite Elements. Applied Mathematical Sciences, Vol. 159, by Alexandre Ern and Jean-Luc Guermond. Springer, 2004.
• Finite Elements : Theory, Fast Solvers, and Applications in Solid Mechanics, by Dietrich Braess, (2nd edition).

  Finite difference methods for partial differential equations, by George E. Forsythe, Wiley, New York, 1960.

  Basic Error Estimates for Elliptic Problems. Handbook of Numerical Analysis, Vol. II, by P.G. Ciarlet, Elsevier, 1991.

Course Description:

This is a one semester course on numerical methods for partial differential equations. The course will focus on the finite element method for elliptic problems but will also include the basic techniques of finite differences and finite volumes. Parabolic and hyperbolic problems will also be considered. The more basic components of the course will be illustrated in computer programming assignments.

MATH 609 and MATH 610 provide background material for the Qualifying Exam in Numerical Analysis . MATH 610 also provides background for the numerical analysis portion of the Combined Applied Analysis/Numerical Analysis Qualifying Exam.

Course Outline:

Elliptic problems.

Variational formulation.

The Ritz-Galerkin method.

Finite element approximation of one dimensional problems.

Numerical integration.

Finite difference methods of one dimensional problems.

Finite volume approximation of one dimensional problems.

Two-dimensional elliptic problems.

Examples of finite elements.

Error estimates and regularity.

Computational considerations.

The finite difference method.

Some non-conforming methods, Strang's Lemmas.

Parabolic problems.

Semi-discrete FEM approximation.

Fully-discrete FEM approximation.

Approximation by finite differences.

Finite volume methods.

Grading Policy:

• 40% will be determined by the homework and programming assignments (assignments turned in late lose 5% per class).
• 60% will be determined by the two exams (25/35)

Prerequisites:

Math 609, Math 641 or a course with some real analysis and basic mathematical skills involving calculus and linear algebra (vector and inner product spaces).

Other information.

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