Math 7760 – Mathematical Foundations of Finite Element Methods, Fall 2007

Jan Mandel

Department of Mathematics
University of Colorado at Denver

This web page http://www-math.cudenver.edu/~jmandel/classes/7760f07 serves as the class syllabus and source of further information. Homeworks, material covered, and handouts are in the class notes file. Photographs and scans will be in the photos directory.

Please check this space before coming to my office or to the class for any last announcements.

 

No class Thursday November 29. Make-up class Monday December 3 11:30-12:45

 

Interesting additional material: Carlos Felippa, History of Matrix Structural Analysis

 

Textbook: Suzanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, 2nd ed, Springer 2002 ISBN 0-387-95451-1

Time and location: Tuesday and Thursday 2:30-3:45 UCD Building Room 641
Office hours: Tuesday and Thursday12-2 UCD building 640
Contact: Phone 303-556-4475; email Jan.Mandel@gmail.com

 

Class objective

Gain understanding of the basic contemporary tools in the theory of partial differential equations and the mathematical analysis of the Finite Element and related methods at a level sufficient to read many current publications in the field.

Catalog description

Theoretical foundations of finite element methods for elliptic boundary value problems, Sobolev spaces, interpolation of Sobolev spaces, variational formulation of elliptic boundary-value problems, basic error estimates, applications to elasticity, practical aspects of the finite element method.

Class approach

Theorem-proof, algorithms, no programming. Homeworks with hard problem solving and assigned independent reading. A good level of mathematical writing will be required. Some complicated proofs will be skipped, but the students will need to be able to expand arguments in deep detail on demand. Discussions with student participation.

 

How to be successful

If you have a problem recalling a result from the prerequisites, review that outside of the class. As always in analysis: question everything, trust nothing. Never take what is in the book or what the instructor says for granted. As what-if questions: what if an assumption in a theorem is not satisfied, would the theorem still hold, would the algorithm still work. Look for motivations: complicated math or algorithms always have a reason – it did not work the simple way; find out why. Study in a group, the best way to learn something is to explain it to others.

Prerequisites

Math 6131 - Real Analysis and its prerequisite Math 5070 - Applied Analysis or equivalent: infimum and supremum, point-set topology in metric spaces, sequences and series of functions, fixed point theorems, Riemann integral, power series, Lebesgue measure and integration, Math 7132 - Functional Analysis: Linear metric and topological spaces, duality, weak topology, spaces of functions, linear operators, compact operators, spectral theory, operator calculus, Math 5663 - Introduction to Finite Element Methods or equivalent programming experience: basic formulation, solution of elliptic and parabolic equations. The prerequisites will be reviewed in the class but such review cannot substitute for a previous in-depth study.

Material to be covered (tentative)

 

  • Ch. 0 – Basic concepts (the 1D case)
  • Ch. 1 – Sobolev spaces
  • Ch. 2 – Variational formulation of elliptic boundary value problems
  • Ch. 3 – The construction of a Finite Element Space
  • Ch. 4 – Polynomial approximation in Sobolev spaces
  • Ch. 5 – n-dimensional variational problems
  • Ch. 6 – Multigrid methods
  • Ch. 7 – Domain decomposition methods
  • Ch. 14 – Operator interpolation theory

Often only selected parts of the chapters will be covered. See the class notes file for the actual progress and homeworks.


Office Hours Policy

I enjoy discussing technical topics related to the class with my students during my office hours or any other time you walk in. However: Please do your homeworks ahead of time, and try not to come to my office on the day a long homework is due and expect me to do it for you.  Please do not try to negotiate grades (except in the case of clear error on my part) or partial credit. I usually need some time before the class to get organized, so please respect the end of the office hours and do not show up right before the class.

Course Policies

 

Attendance will not be recorded. Late homework will be counted 50% if not more than 2 weeks late, after which it will be considered missing and will not count. Homeworks and exams will be usually returned within a week. Because the grading policy allows a 100% score even if a student missed one exam, there will be no make-up exams. Reasonable accommodation (e.g. extension of time for a homework without the penalty if a solution was not posted yet, or a substitute homework) will be made for a good reason such as work demands when requested in advance, and for documented illnesses of the student or a documented family emergency. Cheating and plagiarism or other forms of academic misconduct will result in zero score from the assignment and will be reported to the Department. This includes copying answers from other books without understanding. This is clearly indicated when the student does not adapt the approach and the notation to what is used in this class. Accommodation for disability or military service will be done following current policies in consultation with the Department. Students are expected to follow the Student Code of Conduct. For privacy, graded assignments will be retuned in person in class or the student may provide a self-addressed, stamped envelope. Grades cannot be emailed. Official announcements and other required communication will be made by email to the student’s official University email address. Every student is responsible for reading all email sent to their official address on a timely basis.

Course Schedule, Assignments, Tests, and Grading

There will be homeworks most weeks and in-class midterm and final. The midterm will be on October 18. The final will be on December 11. The homeworks will consist of more challenging problems that will require some independent work. The in-class exams will be concerned with the very basics. The final score will be determined by the sum of the best 2 out out of the following, each scaled separately from 0 to 50: 1. homeworks 2. midterm 3. final. The letter grades will be A >= 90, A- >= 85, B+ >= 80, B >= 75, B- >=70, C+ >= 65, C >= 60, C- >= 55, D+ >= 50, D >= 45, D- >= 40.

Fall 2007 CLAS Academic Policies

 

The following policies pertain to all students and are strictly adhered to by the College of Liberal Arts and Sciences (CLAS).

  • Every student MUST check and verify their schedule prior to the published drop/add deadlines. Failure to verify a schedule is not sufficient reason to justify a late add or drop later in the semester. It is the student’s responsibility to make sure that their schedule is correct prior to the appropriate deadlines.
  • CLAS students must use their email.cudenver.edu email address. Email is the official method of communication for all University of Colorado at Denver and Health Sciences Center business. All email correspondence will take place using your UCDHSC email address. Go to http://www.cudenver.edu/registrar  to activate your email.cudenver.edu email address.
  • Students are NOT automatically added to a course off a wait list after wait lists are dropped. If a student is told by a faculty member that they will be added off the wait list, it is the responsibility of the student to complete the proper paperwork to add a course. Students are NOT automatically added to a course off the wait list after the 5th day of the semester when wait lists are dropped.
  • Students are not automatically notified if they are added to a class from a wait-list. Again, it is the responsibility of the student to verify their schedule prior to any official dates to drop or add courses.
  • Students must complete and submit a drop/add form to make any schedule changes. Students are not automatically dropped from a class if they never attended, stopped attending or do not make tuition payments.
  • Late adds will be approved only when circumstances surrounding the late add are beyond the student’s control and can be documented independently. This will require a petition and documentation from the student. Late adds will only be approved if the student has not taken any exams, quizzes, or has not completed any other graded assignments. Independent verification of this from the professor of record will be required. Please note that the signature of a faculty member on an add form does not guarantee that a late add petition will be approved. Petitions are available in NC 2024.
  • Late drops will be approved only when circumstances surrounding the late drop are beyond the student’s control and can be documented independently. This will require a petition and documentation from the student. Please note that the signature of a faculty member does not guarantee that a late drop petition will be approved. Petitions are available in NC 2024.
  • Students wishing to graduate in fall of 2007 must meet with their academic advisor by the end of the drop/add period to obtain a graduation application. This application must be completed and submitted by 5 PM on September 5, 2007. You can obtain an application ONLY after meeting with your academic advisor. There are no exceptions to this policy or date.
  • Students are responsible for completing financial arrangements with financial aid, family, scholarships, etc. to pay their tuition. Students will be responsible for all tuition and fees for courses they do not officially drop using proper drop/add procedures and forms.
  • Students who drop after the published drop/add period will not be eligible for a refund of the COF hours or tuition.

 

 

 

 

Important Dates

 

  • August 20, 2007; First day of Class
  • August 26, 2007; Last day to be added to a wait list.
  • August 26, 2007: Last day to add a course using the SMART system.
  • August 20-August 27, 2007; Students are responsible for verifying an accurate fall 2007 course schedule via the SMART registration system. Students are NOT notified of their wait-list status by the university. All students must check their scheduled prior to August 27, 2007 for accuracy.
  • August 27, 2007: LAST DAY TO DROP WITHOUT DROP CHARGE.
  • August 27, 2007: Wait Lists are dropped. Any student who was not added to a course automatically from the wait list by this date and time MUST complete a drop/add form to be added to the class. Students are NOT automatically added to the class from the wait list after this date and time. If your name is not on the official student roster, you are not registered for the course.
  • August 28, 2007: First day instructor may approve request to add a student to a full course with a Schedule Adjustment Form.
  • August 31 – September 5, 2007: Drops allowed using SMART system. Drop charge applies. Full term courses may be added using Schedule Adjustment Form.
  • September 5, 2007 at 5 PM; Last day to add structured courses without a written petition for a late add. This is an absolute deadline and is treated as such. This deadline does not apply to independent study, internships, and late-starting modular courses.
  • September 5, 2007 at 5 PM; Last day to drop a fall 2007 course with a tuition refund minus the drop charge and no transcript notation. Drops after this date will appear on your transcript. This is an absolute deadline and is treated as such.
  • September 5, 2007 at 5 PM; Last day to completely withdraw from all fall 2007 courses with a tuition refund and no transcript notation. Drop charge applies. Drops after this date will appear on your transcript. This is an absolute deadline and is treated as such.
  • September 5, 2007 at 5 PM; Last day to request pass/fail option for a course.
  • September 5, 2007 at 5 PM: Last day to request a no credit option for a course.
  • September 5, 2007 at 5 PM: Last day to register for a Candidate for Degree.
  • September 5, 2007 at 5 PM: Last day to petition for a reduction in thesis or dissertation hours.
  • September 5, 2007 at 5 PM: Last day to apply for fall 2007 graduation. You must make an appointment and see your academic advisor to apply for graduation.
  • After September 5, 2007 all schedule changes require a petition. Petitions are available in NC 2024.
  • October 29, 2007 at 5 PM; Last day for non CLAS students to drop or withdraw from all classes without a petition and special approval from the student’s academic Dean. This is treated as an absolute deadline.
  • November 9, 2007 at 5 PM; Last day for CLAS students to drop or withdraw from all classes without a petition and special approval from the student’s academic Dean. This is treated as an absolute deadline.
  • No schedule changes will be granted once finals week has started. There are NO exceptions to this policy.