MATH 661 - 600: Mathematical theory of finite element methods, Fall 2003

General information

• Instructor: Dr. Raytcho Lazarov, Blocker 505C, 845 7578
• Time: MWF 9:10 -- 10:00 am
• Classroom: Blocker 156
• Office Hours: MWF 1:30 - 2:30 pm or by appointment
• Text: (Required) S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, 1994.

Course description

• This is a one-semester course in the general area of numerical analysis. Theoretical aspects of the finite element (FE) method for elliptic and parabolic partial differential equations will be discussed. The first part of course will follow closely the textbook. The second part will cover special topics such as mixed method, least-squares, a postriori error analysis, etc.
• The prerequisites are one semester of numerical analysis for PDEs (something equivalent to Math 610) and knowledge of a programming language.

Course Outline

• Basic concepts: weak formulation of BVPs, Ritz-Galerkin approximation, finite element method, Petrov-Galerkin method, finite volume element method.
• Sobolev spaces and variational formulation of elliptic and parabolic BVPs: Sobolev norms and spaces, trace theorem, Hilbert spaces, Riesz representation theorem, estimates for general finite element approximations.
• The construction of finite elemenet spaces: triangular FE, rectangular FE, higher dimensional elements and exotic elements.
• Polynomial approximation theory in Sobolev sapces (Bramble-Hilbert lemma): avaraged Taylor polynomials and bounds for Riesz potentials, bounds for the interpolation error, isoparametric polynomial approximations.
• Variational crimes due to: (1) nonconforming FE, (2) approximation of the domain, (3) isoparametric FE and quadratures.
• Special topics (mixed methods, least-squares, a postriori error analysis, discontinuous Galerkin method, if time permits).

Grading Policy

• Your grade for the course will be computed as follows:
  • (a) 25% will be determined by your programming assignments/projects
  • (b) 50% will be determined by your two exams (the final is comprehensive)
  • (c) 25% will be determined by your homework assignments
• your MINIMUM grade will be A, B, C, or D, for averages of 90%, 80%, 65%, or 50%, respectively.

Some supplementary books you might find useful

Introductory Texts in Numerical PDEs:

• A. Tveito and R. Winther, Introduction to Partial Differential Equations; A computational Approach, Springer, 1998;
• C. Johnson, Numerical Solution of PDEs by the Finite Element method, Cambridge University Press, 1995.
• R. Kress, Numerical Analysis, Springer, 1998.

Advanced Theoretical Texts in Numerical PDEs:

• D. Braess, Finite Elements; Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 1997.
• P. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, 2002.
• L.B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Springer, 1991.

Computational Issues and Engineering Application:

• O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, Vol. 1: The Basis, Butterworth-Heinemann, 2000.
• O.C. Zienkiewicz and R.L. Taylor, The FInite Element Method, Vol. 2: Solid Mechanics, Butterworth-Heinemann, 2000.
• O.C. Zienkiewicz and R.L. Taylor, The FInite Element Method, Vol. 3: Fluid Dynamics, Butterworth-Heinemann, 2000.