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Mathematics 16:642:575 Numerical Solution of Partial Differential Equations

Schedule

The course is usually offered every two years during the Spring semester.
  • Class meeting dates: Please visit the University's academic calendar.
  • Schedule and Instructor: Please visit the University's schedule of classes for the instructor, time, and room.
  • Instructor and Teaching Assistant Office Hours: Please visit the Mathematical Finance program's office hour schedule.

Course Abstract

In this course, we study finite difference, finite element, and finite volume methods for the numerical solution of elliptic, parabolic, and hyperbolic partial differential equations and variational inequalities. The course will concentrate on the key ideas underlying the derivation of numerical schemes and a study of their stability and accuracy. Students will have the opportunity to gain computational experience with numerical methods with a minimal of programming by the use of various software packages.

Pre-requisites

At least one of Numerical Analysis I (16:642:573) or Numerical Analysis II (16:642:574), or permission of the instructor.

Primary Textbooks

Note: Since detailed lecture notes will be available on the course web site, there is no assigned textbook. However, since students often find a textbook useful, the following three books are recommended, all available in paperback.

Dietrich Braess, Finite Elements: Theory, fast solvers, and applications in solid mechanics, 3rd ed., Cambridge University, 2007.

Stig Larsson and Vidar Thomée, Partial Differential Equations with Numerical Methods, Texts In Applied Mathematics, Volume 45, Springer, 2009.

Claes Johnson, Numerical Solutions of Partial Differential Equations by the Finite Element Method, Dover Books on Mathematics, 2009.

Grading

Please contact the instructor.

Class Policies

Please see the MSMF common class policies.

Assignments

Homework assignments in the course consist of both theoretical and computational work. The computational assignments can be done using Matlab or one of the special packages for solving partial differential equations to be chosen by the instructor. There will be one assignment for each 3-4 class periods. Since solutions will be posted to the course web site, late homework assignments pose a problem. Students with exceptional circumstances may be granted short extensions. Please contact the instructor as soon as a problem arises.

Previous Instructor Course Websites

2010 Young-Ju Lee
2012 Richard Falk

Weekly Lecturing Agenda and Readings

The lecture schedule below is a sample; actual content may vary depending on the instructor.

Lecture Topics
1 Finite Difference Methods for Elliptic Problems
2 Stability and Error Estimates
3 Extensions of the Method
4 Finite Element Method for Elliptic Equations - Introduction
5 Finite Element Method for Elliptic Equation
6 Construction of finite element subspaces
7 Affine families of finite elements
8 Error estimates for piecewise linear interpolation
9 Error estimates by scaling
10 Order of Convergence and other Finite Elements
11 Approximation of saddle point problems
12 Error estimates for the approximation of saddle point problems
13 Application to the mixed finite element method for Poisson's equation
14 Application to the stationary Stokes equations
15 Efficient solution of the linear systems arising from finite element discretization
16 Efficient solution of the linear systems arising from finite element discretization
17 Finite difference methods for the heat equation
18 Finite difference methods for the transport equation and the wave equation
19 Stability of difference schemes for pure IVP with periodic intial data
20 Stability of difference schemes -- examples
21 Qualitative properties of finite difference schemes
22 Finite element methods for parabolic problems
23 A finite element method for the transport equation
24 Approximation of hyperbolic conservation laws

Library Reserves

All textbooks referenced on this page should be on reserve in the Hill Center Mathematical Sciences Library (1st floor). Please contact the instructor if reserve copies are insufficient or unavailable.