Syllabus: MTH 652, Winter 2019

Advanced Numerical Analysis II

Instructor: Jay Gopalakrishnan
Times: Tue, Thu: 08:30-09:45
Venue: HAR 104 (Harder House)
Office hours: HAR 105, 10:00-11:00

Learning Outcomes

This course provides a mathematical introduction to finite elements. At the end of the course the students would have gained familiarity with algorithms for numerically solve boundary values problems on complex domains, software tools implementing the algorithms, and functional analytic tools for analyzing the approximation of solutions.


MTH 651 of Fall 2018, or permission of the instructor.

Topical outline

Closed range theorem. Babuska-Brezzi theory. Lax-Milgram and other simplified cases.
Examples. Dirichlet problem. Linear elasticity. Maxwell equations. Stokes flow.
Approximation in Sobolev spaces.
Some aspects of design of preconditioners.
Various methods for Stokes flow.
Various methods for linear elasticity.
Good and bad methods for Maxwell equations.

(For a more detailed daily schedule, please see the class diary, after logging to D2L.)

Learning methods

Theory: Most of the mathematical theory will be covered in lectures. Exercises will be given throughout. Some of the meeting times will be devoted to solving exercise problems.

Code: In this course, we shall use Python 3 and its SciPy and NumPy modules. For studying practical aspects of finite elements, we will use the open source software NGSolve.

Some meeting times will reserved for hands-on software sessions. Students are expected to bring their own laptops for these sessions. Enough time was spent last term on introduction to, and installation of, NGSolve and other python modules, so its expected that all students already have ready access to these within their preferred computing environment.

References: There is no required textbook for this course. These reference books will be helpful:

  • Theory and Practice of Finite Elements, by A. Ern and J.-L. Guermond, Springer, 2004.
  • The Mathematical Theory of Finite Element Methods, by Susanne Brenner and Ridgway Scott, 3rd edition, Springer, 2008.
  • The finite element method for elliptic problems, by Philippe G. Ciarlet, 2nd edition, SIAM's 2002 reprinting of classics.
  • Finite element methods for Maxwell's equations, by Peter Monk, Oxford University Press, 2003.

Course Management System: We will use the university's D2L, where (after logging in) this course's landing page can be found. All course materials will be placed there.

Evaluation of learning

Grades will be assigned based on take-home exams/projects.

Fine print

Academic Misconduct: In the list of prohibited student behavior at PSU is plagiarism, buying and selling of course assignments, and obstruction of another student's success. Students are expected to know of and refrain from all proscribed conduct.
Title IX Reporting Obligations: Every instructor at PSU has the responsibility to help create a safe learning environment for students and for the campus as a whole. As a member of the university community, an instructor must report any instances of sexual harassment, sexual violence and/or other forms of prohibited discrimination. If you would rather share information about sexual harassment, sexual violence or discrimination to a confidential employee who does not have this reporting responsibility, please use the online list of those individuals. For more information about Title IX please complete the student module Creating a Safe Campus in D2L.
Disability Accommodations: The Disability Resource Center (DRC) provides reasonable accommodations for students who encounter barriers in the learning environment. If you have, or think you may have, a disability that may affect your work in this class and feel you need accommodations, contact the DRC to schedule an appointment and initiate a conversation about reasonable accommodations. The DRC is located in 116 Smith Memorial Student Union, 503-725-4150,, Students who have testing accommodations must begin the test at the same time as the rest of the class.
Course Materials: All course materials handed out in class or placed in D2L are solely for the use of each student registered in this course. Sale of these materials is prohibited. During class sessions, voice or video recording of the instructor or other students without their explicit written consent is prohibited.

Author: Jay Gopalakrishnan

Last updated: 2019-01-08 Tue 07:17