Math652: Finite Element Methods - Colorado State University

Department of Mathematics

Colorado State University

Math652: Finite Element Methods

Classroom: Instructor:

Office: Office Hours: Course Page:

ENGRG 206

Time: MTWF 3:00-3:50PM

Yongcheng Zhou Phone: 491-0237

124 Weber

Email: yzhou@math.colostate.edu

M 8:30-9:30AM, W: 10:00-11:00AM, F: 10:00-11:00AM or By appointment

spring2009/

Textbook Partial Differential Equations with Numerical Methods, by Stig Larsson and Vidar Thom?ee. Course description This course is on the analysis and computer implementation of numerical methods for partial differential equations, with a focus on finite element methods. The topics include the basic ideas of finite difference methods, finite volume methods and finite element methods, functional spaces for numerical approximation, weak formulation of PDEs, various error estimates and adaptivity. Students will get access to general framework of numerical treatment of PDEs, study indispensable techniques for error estimation and convergence analysis, and gain hand on programming experiences in computer implementation of their own numerical methods. Prerequisites Knowledge of the following topics is required:

? Numerical Analysis. (Math 450/451 or Math 561)

? Partial differential equations and linear algebra. (Math 332 or Math 546)

? Computer programming (MATLAB, Fortran or C/C++)

Homework Homework is assigned in class (and posted on the class webpage) every Friday. The problems involve a mix of theory and computing. Regarding the latter, please read the text below the Programming heading. Most of the homework problems are from the textbook.

Your submitted homework should show all necessary work you used to solve the problems; mathematical statements should be complete (or nearly complete) sentences; and the reasoning and logic underlying all arguments should be clearly spelled out. Computer programs should be included along with numerical results presented in a readable format (e.g., in a table with heading or in a plot with labels). Failure to adhere to the above requirements may result in a loss of points.

Homework assignments are due on Friday at the end of class. Late homework is not accepted unless arrangements are made in advance of the deadline. Teamwork Teamwork is part of the real world and therefore permitted (and encourage) for all programming assignments excepts the exams. The purpose of teamwork is to enhance the learning effect, rather than to decrease the workload. It is up to each team member to prevent abuse. Please observe the following important rules:

? A team should have no more than three members

? If you work in a team then clearly state so and hand in just on set of answers. If you collaborate on some problems but not on others then clearly state so on your turned-in solution

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Department of Mathematics

Colorado State University

? In general, you can consult literature and people, but you have to acknowledge all help so obtained except for the textbook and the course instructor

Homework solutions (taken from selected students' solutions) are posted on the course webpage once the grading has concluded. Grading Policy The final grade for the course is based on homework assignments, a midterm exam, and a final exam. The breakdown for the course grade is as follows:

? Homework: 70%

? Final: 30% (Final Projects and Oral Presentations)

There are more than 7 homework assignments and we only count the highest 7 scores. The grade of the course is calculated as follows:

A: 100%-90%; B: 89%-80%; C: 79%-70%; D: 69%-60%; F: 59% or below Programming Students are highly recommended to use MATLAB for all programming assignments involving numerical computations. In order to gain exposure to numerical computation, students need to practice programming and using software packages. MATLAB offers the perfect opportunity for such practice; it is one of the most dominant commercial computing environments. The MATLAB language is intended to be easy to learn and use, while still being extremely powerful. Three other reasons for using MATLAB for the programming assignments is that: (1) the webpage for the book contains a MATLAB implementation of several algorithms discussed; (2) My examples and homework assignments will use MATLAB code; (3) There are many mini MATLAB packages available for quickly implementing finite element methods; (4) I will help you debug your MATLAB programs (scripts).

If you have not used MATLAB previously, help resources are available on the course webpage and from the instructor. MATLAB is available in the Undergraduate and Graduate student mathematics computing labs (as well as various other locations on campus). You can also purchase a student version of MATLAB for a heavily discounted price for the book store.

References

? Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, Springer

? Roberts A. Adams, Sobolev Spaces, Academic Press (2nd Edition)

? Lawrence C. Evans, Partial Differential Equations, AMS

? Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM (2nd Edition)

? Alexandre Ern and Jean-Luc Guermond, Theory and Practice of Finite Elements, Springer

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