Math 3072: Finite Element Method

Instructor: Catalin Trenchea
Phone: (412) 624-5681
Office: Thackeray 606

Lecture: MWF 10:00-10:50 THACKERAY 524

Office Hours: Tu 10:00am-noon, Thu 10:00am-noon and by appointment

Textbook: The course will not follow closely a specific text book. The book Understanding and Implementing the Finite Element Method, by Mark Gockenbach, SIAM 2006, will be used as a reference on implementation issues.

Prerequisites: Good undergraduate background in linear algebra and advanced calculus. Familiarity with partial differential equations will be useful.

Content: This course is an introduction to the theoretical and computational aspects of the finite element method for the solution of boundary value problems for partial differential equations. Emphasis will be on linear elliptic, self-adjoint, second-order problems, and some material will cover time dependent problems as well as nonlinear problems. Topics include: Sobolev spaces, variational formulation of boundary value problems, natural and essential boundary conditions, Lax-Milgram lemma, approximation theory, error estimates, element construction, continuous, discontinuous, and mixed finite element methods, and solution methods for the resulting finite element systems.

Topics to be covered: FEM for two point boundary value problems, Brief introduction to Sobolev Spaces, FEM for elliptic equations, Approximation theory for FEM, Non-conforming and mixed FEM, FEM for parabolic equations, FEM for advection diffusion equations, FEM for hyperbolic equations.

Homework: Written homework and several computational projects will be assigned. Suggested programming languages are FreeFEM and Matlab. In the computational projects we will utilize the software provided with the textbook.

Exams: There will be one in-class midterm exam, a paper presentation, and a final project.

Other references:

The mathematical theory of finite element methods, S. C. Brenner and L. R. Scott, Springer-Verlag 1994.

Finite elements, 2nd ed., D. Braess, Cambridge 2001.

The Finite Element Method for Elliptic Problems, Philippe G. Ciarlet, SIAM 2002.

Numerical Solutions of Partial Differential Equations by FEM, Claes Johnson, Studentlitteratur 1987.

Finite element solution of boundary value problems, O. Axelsson and V. A. Barker, SIAM 2001.

The course web page will be updated continuously throughout the semester. The student is responsible for checking this web page for assignements and policies.

If you have a disability for which you are or may be requesting an accodomation, you are encouraged to contact both your instructor and the Office of Disability Resources and Services, 216 William Pitt Union, (412) 648-7890/(412) 383-7355 (TTY), as early as possible in the term. DRS will verify your disability and determine reasonable accomodations for this course.