Math 7824 Topics in Computational Mathematics

Numerical Methods for Partial Differential Equations

Spring 2010

 

Instructor: Jan Mandel, room CUD 640, 303-556-4475

Web: http://math.ucdenver.edu/~jmandel/classes/7824s10

Text: 1. C. Johnson,  Numerical solution of partial differential equations by the finite element method,  Dover, 2009 ISBN 978-0486469003 2. R. J. LeVeque, Finite Difference Methods for Ordinary and partial Differential Equations, SIAM 2007 ISBN 978-0-898716-29-0

Time and place: Mo-We 10:00-11:15 room CUD 641

Office hours: Mo-We 8:45-9:45 room CUD 640. I am available after the class also except when there is a CCM Colloquium or a faculty meeting.

Communication: I am best reached by email at jan.mandel@gmail.com. The material covered and homework will be in the notes file. The grades will be recorded on Blackboard. I take photographs of all blackboards, which are stored in the photos directory.

Approach:  Mostly practical and algorithm oriented, with rigorous analysis only when a simple analysis is feasible. Otherwise, the analysis will often rely on approximate arguments and computational experiments.

Purpose: Basics of the finite element methods as generally expected of students in computational mathematics who do not specialize in finite elements. Finite difference methods as used in practical finite difference codes. Prerequisite methods for ODEs are included but in less detail than usually covered in Numerical analysis II and Numerical ODE courses.

Description:  Solving partial differential equations numerically is one of the most frequent applications of mathematics and computing in science and engineering, and it constitutes the vast majority of applications running on supercomputers. This class will provide an introduction into numerical methods for stationary and time-dependent problems from a mathematical perspective. It is suitable as a complement to a finite elements engineering course, the first course in the subject for a numerical analyst, as well as an overview for mathematics and engineering graduate students.
Prerequisites: Working knowledge of normed vector spaces and some computer programming experience. Previous courses in analysis, numerical analysis, or partial differential equations, are helpful but not required.
Tentative topics: Finite element methods - variational formulation of  Laplace equation, Sobolev spaces, basic error estimates, construction of finite element spaces,  approximation properties, implementation, sparse matrices, applications to elasticity and structures. Finite difference methods - 5-point scheme for the Laplace equation, truncation error, solution by Fast Fourier Transform, methods for time-dependent problems (Euler, Ruge-Kutta,  Crank-Nicholson, leapfrog),  A-stability,  heat equation, nonlinear problems, consistence, stability, convergence, methods for advection problems, upwinding,  applications to fluid dynamics and level set methods.

Grading: 1/3 for 3 best from the following 4: homework, midterm, final, project. There will be homework, theoretically or practically oriented, most weeks. You will always have at least a week to complete the homework. The midterm and the final are closed book, no notes except for one handwritten letter-sized sheet (both sides). The midterm will be on March 17. The final will be during the finals week as determined by the Auraria schedule. The letter grades are:

A >=90%, A->=85%, B+>=80%, B>=75%, B->=70, C>=55%, D>=40%.

Technology: You are required to use Matlab for computing and submit all computer assignments and the project including your presentation online, using a system I will provide. Please do not send any files by email. You are encouraged to scan or typeset the theoretical assignments and submit them online as well. You can use your own computer if you wish, or you can use Matlab on the math server. You can get full credit for a computer assignment only if your code runs as submitted, and only if the results are correct. IÕll be happy to provide advice, but I will not fix or debug your code.

Policies: Class attendance is recommended. Late homework gets half credit. Because the worst part of the grade is dropped, there are no makeups. Cheating or other academic misconduct will result in no credit for the assignment or exam in question, plus lowering the grade by 10 % points, and will be reported to the department chair.

 

Spring 2010 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 studentÕs responsibility to make sure that their schedule is correct prior to the appropriate deadlines.
  • CLAS students must use their email.ucdenver.edu email address.  Email is the official method of communication for all University of Colorado Denver business.  All email correspondence will take place using your UCDHSC email address.  Go to http://www.ucdenver.edu/student-services/resources/registrar/students/policies/Pages/EmailPolicy.aspx to activate your 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 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 studentÕs 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 on an add form does not guarantee that a late add petition will be approved.  Petitions are available in NC 4011.
  • Late drops will be approved only when circumstances surrounding the late drop have arisen after the published drop deadlines, are beyond the studentÕs control, and can be documented independently.  This will require a petition and documentation from the student.  Pre-existing circumstances (circumstances that existed prior to the published drop deadlines) regarding illness, work, family, or other confounding issues will not be considered adequate reason to drop or withdraw from courses after the published University and/or College drop deadlines.  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 4011.
  • Undergraduate students wishing to graduate in spring of 2010 must meet with their academic advisor by census date to obtain a graduation application.  This application must be completed and submitted by 5 PM on February 3, 2010.  You can obtain an application ONLY after meeting with your academic advisor. There are no exceptions to this policy or date.
  • Graduate students wishing to graduate in spring semester 2010 must complete their Intent to Graduate form and have a Request for Admissions to Candidacy on file with the CLAS DeanÕs office no later than 5 PM, February 3, 2010.
  • 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

 

  • January 19, 2010: First day of Class
  • January 24, 2010: Last day to add a class or be added to a wait list for a class using the SMART system.
  • January 25, 2010: LAST DAY TO DROP WITHOUT DROP CHARGE – THIS INCLUDES SECTION CHANGES.
  • January 25, 2010: 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 schedule adjustment 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.
  • January 26-February 3, 2010: Students are responsible for verifying an accurate spring 2010 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 February 3, 2010 for accuracy.
  • January 26, 2010: First day instructor may approve request to add a student to a full course with a Schedule Adjustment Form.
  • February 3, 2010: Census date.
  • February 3, 2010 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, project hours, thesis hours, dissertation hours, and late-starting modular courses.
  • February 3, 2010 at 5 PM: Last day to drop a spring 2010 course or completely with draw from all spring 2010 courses using a schedule adjustment form with a tuition adjustment minus the drop charge and no transcript notation – this includes section changes.  Drops after this date will appear on your transcript.  This is an absolute deadline and is treated as such.
  • February 3, 2010 at 5 PM: Last day to request pass/fail or no credit option for a course.
  • February 3, 2010 at 5 PM: Last day to for a graduate student to register for a Candidate for Degree.
  • February 3, 2010 at 5 PM: Last day for a Ph.D. student to petition for a reduction in hours.
  • February 3, 2010 at 5 PM: Last day to apply for spring 2010 graduation.  You must make an appointment and see your academic advisor before this date to apply for graduation if you are an undergraduate; you must complete the intent to graduate and candidate for degree form if you are a graduate student.
  • February 15-24, 2010: Faculty can use the early alert system.
  • March 22-28, 2010: Spring break (no classes/campus open)
  • April 2, 2010 at 5 PM: Last day for non CLAS students to drop or withdraw from all classes without a petition and special approval from the studentÕs academic Dean.  After this date, a deanÕs signature is needed.
  • April 16, 2010 at 5 PM: Last day for CLAS students to drop or withdraw from all classes with signatures from the faculty and Dean without a petition.  This is treated as an absolute deadline.
  • After April 16, 2010 all schedule changes require a full petition.  Petitions are available in NC 4011.
  • May 10-15, 2010: Finals Week
  • No schedule changes will be granted once finals week has started.  There are NO exceptions to this policy.