COURSE SYLLABUS

Course Number: MATH7610
Course Title: NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 
Credit Hours: 3
Prerequisites: MATH 6640 or departmental approval.
Corequisite: 

Objectives: 
The numerical solution of partial differential equations using finite difference and finite element methods.
The course is an introduction to the finite element and finite difference methods for solving boundary-value partial differential equations and initial-value, boundary-value, partial differential equations.
Included is discussion of stability consistency and convergence of finite difference schemes, and see error estimates for the finite element method. The students will also write numerical code to solve simple partial differential equations in order to get ``hands on'' experience with these methods.

Course Content (typical):

Introduction: (2 weeks)
The Finite Difference Method., The Finite Element Method.  Finite Differences vs. Finite Elements.
Review of Partial Differential Equations.
Elliptic equations, parabolic equations hyperbolic equations.
Examples of some equations: the vibrating string, heat conduction in an isotropic solid, elastic deformation, fluid flow.

Finite Differences: (4 weeks)
Parabolic Problems. Explicit schemes, implicit schemes, treatment of boundary conditions, Lax equivalence theorem, Fourier analysis, stability, consistency, convergence. 
Hyperbolic Problems. Method of characteristics, initial and boundary conditions for hyperbolic equations, stability condition (Courant-Friedrichs-Lewy condition), explicit schemes, the analytic interval of dependence and the numerical interval of dependence, implicit schemes, stability, consistency, convergence.

Finite Elements:} (8 weeks)
Elliptic Problems. Variational formulation, weak formulation, discretization, discrete systems of equations, typical error estimates, introduction to Hilbert spaces and Sobolev spaces, Lax Milgram Lemma, Dirichlet Neumann and Mixed problems, error estimates, programing and implementation, remarks on adaptive methods.
Parabolic Problems. Weak formulation, semidiscretization, time discretization explicit and implicit, discontinuous Galerkin method, error estimates.

Other Topics: as time permits (1 weeks)
Numerical Methods for Large Sparse Systems of Equations, Sparse storage, iterative methods, Conjugate Gradients.
The Spectral Method. The Boundary Element Method.

Possible Texts:
D.Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, Cambridge, 1997.
K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University
Press, Cambridge, 1994. 

Sample Grading and Evaluation Procedures

A graduate student is expected to creatively engage the mathematical material of the course.  The student will be given problems to solve; these problems may include the derivation of proofs to theorems.  These solutions may be presented in class on the blackboard or be written up to be handed in to the instructor. Extended projects may also be assigned.  

Grade Calculation
Presentation of solutions to problems/theorems, homework:				40%
Midterm exam or midterm project:							25%
Final Exam or culminating project:							35%

There may be variations in these procedures depending on the individual instructors and the nature of the specific material.
Sample Statement Re: Accommodations
	Students who need accommodations are asked to arrange a meeting during office hours the first week of classes, or as soon as possible if accommodations are needed immediately.  If you have a conflict with my office hours, an alternate time can, be arranged.  To set up this meeting, please contact me by E-mail.  Bring a Copy of your Accommodation Memo and an Instructor Verification Form to the meeting.  If you do not have an Accommodation Memo but need accommodations, make an appointment with The Program for Students with Disabilities, 1244 Haley Center, 844-2096 (V/TT).  (Note: Instructor office room, office hours and email address will be made available on the course syllabus and on the first day of class.)

JUSTIFICATION FOR GRADUATE CREDIT

	This course is part of a modified semester version of a 600-level course under the quarter system.  Under the quarter system it was specifically designed as a graduate course.  It was approved as part of our graduate program and has been a traditional part of our graduate program offerings.  Outside of modernization, the standard of the course remains at the same graduate level that the department has maintained in the past. The course demands considerable mathematics background and a degree of mathematical maturity traditionally found at the graduate level.  The 7000-level course will inculcate the same analytical skills and depth of understanding previously demanded by the comparable 600-level quarter course.  In order to successfully complete the course the student will have to demonstrate an ability to creatively examine and apply the mathematics presented in the course.