MATH610-600: Numerical Methods for PDE's, Spring 2012

General information

• Instructor: Dr. Raytcho Lazarov, Blocker 505C
• Teaching Assistants: Youli Mao , Blocker 505
• Time: TR 11:10 -- 12:25 (regular class), W 9:10 -- 10:00 am computer lab
• Classroom: Block 148 (regular class), BLOC 123 computer lab
• Office Hours: TR 10:00 -- 11:00 am or by appointment
• Texts: (basic) Numerical Treatment of Partial Differential Equations, by C. Grossmann, H.-O. Ross, and M. Stynes, Springer, 2007, and (additional) Theory and practice of finite elements, by A. Ern and J-L Guermond, Springer, 2004.

Course description

• This is a one-semester course on numerical methods for partial differential equations. The course will focus on the basic concepts of the finite element method for elliptic boundary value problems. These will include a weak (variational) formulation of the problem and relies on the coercivity and continuity of certain bilinear forms in the simplest setting (and inf-sup condition in more general cases), approximation theory by piece-wise polynomial functions over partition of the domain into finite elements, error analysis, stability, and variational "crimes".
• Further, some basic techniques of finite differences and finite volumes will be introduced and discussed. Maximum principle, energy type estimates, and Fourier mode analysis will be used for studying the stability and accuracy of these methods applied to for elliptic, parabolic and hyperbolic problems.
• The intention is to develop fairly stong mathematical knowledge and expertise so after completing the class the students can read and understand quite advances papers on finite element methods for PDEs (and partially for finite differences, finite volumes). The programming assignments will emphasize on the applications of the numerical techniques to engineering problems.
• This course covers entirely the numerical analysis part of the qualifying exam in ApplMath/NA.

Practical Issues

• Graphics, grid generation, program implementation and simple iterative techniques will be discussed (mostly in the labs) in order to help you with the programming assignments. Programming skills (e.g. C, C++ or Matlab) are essential.
• Prerequisite for this class are MATH 417 or 609 or equivalent. However, a basic knowledge of PDEs or engineering models that lead to boundary and initial value problems are essential. A good basis for this is Math602.
• The students from engineering departments are encouraged to make their own project that is related to either to their research funded by the corresponding graduate program or to their personal interests and/or preferences. Such project should be assigned by the end of March and completed by the end of April. It could be used to replace one of the programming assignments.

Exam Schedule

• Test #1 , Tuesday, February 14, 2012,
• Test #2, Tuesday, April 3, 2012,
• Test # 3, Wednesday, May 2, 2012, Blocker 164, 5:30 -- 7:45 pm

Grading Policy

• Your grade for the course will be computed as follows:
  • (a) 20% will be determined by the programming assignments ;
  • (b) 50% will be determined by the three tests (T#1 - 15%, T#2 - 15%, and Final#T - 30 %; the final is comprehensive);
  • (c) 20% will be determined by the homework assignments ;
• your MINIMUM grade will be A, B, C, or D, for averages of 90%, 75%, 60%, or 50%, respectively.