Mathematics 16:642:575 Numerical Solution of Partial Differential
The course is usually offered every two years during the Spring
- Class meeting dates:
Please visit the University's academic
- Schedule and
visit the University's schedule
of classes for the instructor, time, and room.
and Teaching Assistant Office Hours: Please visit the
Mathematical Finance program's office
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.
At least one of
or permission of the instructor.
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.
Please contact the instructor.
Please see the MSMF common
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
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.
|| Finite Difference Methods for Elliptic Problems
|| Stability and Error Estimates
|| Extensions of the Method
|| Finite Element Method for Elliptic Equations - Introduction
|| Finite Element Method for Elliptic Equation
|| Construction of finite element subspaces
|| Affine families of finite elements
|| Error estimates for piecewise linear interpolation
|| Error estimates by scaling
|| Order of Convergence and other Finite Elements
||Approximation of saddle point problems
|| Error estimates for the approximation of saddle point
|| Application to the mixed finite element method for Poisson's
|| Application to the stationary Stokes equations
|| Efficient solution of the linear systems arising from finite
||Efficient solution of the linear systems arising from finite
|| Finite difference methods for the heat equation
|| Finite difference methods for the transport equation and the
|| Stability of difference schemes for pure IVP with periodic
|| Stability of difference schemes -- examples
|| Qualitative properties of finite difference schemes
|| Finite element methods for parabolic problems
|| A finite element method for the transport equation
|| Approximation of hyperbolic conservation laws
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.