Numerical Analysis of Differential Equations 1st semester
|Fall 2012, MWF 10:10-11:00 Vincent Hall 207|
|Instructor: Douglas N. Arnold|
|Contact info: 512 Vincent Hall, tel: 6-9137, email:
|Office hours: Monday 3:30-4:20, Wednesday 2:30-3:20, and by appointment|
About the course: This is the first semester of a two-semester graduate level
introduction to the numerical solution of partial differential equations.
In the first semester
will begin with finite difference methods for the Laplacian
and the basic techniques to analyze them (maximum principle, Fourier analysis,
energy estimates). It will then continue with a study of
numerical linear algebra relevant to solution of discretized PDEs, such as those arising
from the finite difference discretization of the Laplacian (classical iterations,
conjugate gradients, multigrid). The largest portion of the
first semester will be devoted to finite element methods for elliptic problems,
and their analysis. The semester will conclude with the use of finite difference
and finite element methods to solve time-dependent problems. The course will include
computational examples and projects using
Matlab, and, especially, the FEniCS software suite. A feature of the course is that
we will emphasize a uniform framework based on consistency and stability to analyze
both finite element and finite difference methods, for both stationary and
The cost for inadequate numerical analysis can be high. The first time
this offshore platform was installed, it
crashed to the sea bottom causing a seismic event measuring 3.0 on the
Richter scale and costing $700,000,000. The cause: flawed algorithms for
the numerical solution of the relevant partial differential equations.
For more information see here.
Text and syllabus:
The course will follows these Lecture Notes.
The table of contents may be taken as the syllabus for the course.
Similar material is covered
in the following texts. These are all on reserve in the
Math Library and several have electronic editions available
through the library.
- Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems, Randall J. Leveque, Society for Industrial and Applied Mathematics, 2007.
- Finite elements: theory, fast solvers, and applications in solid mechanics, Dietrich Braess, third edition, Cambridge University Press, 2007.
- The finite element method for elliptic problems, Philippe Ciarlet, Elsevier North-Holland, 1978.
- Numerical solution of partial differential equations by the finite element method, Claes Johnson, Cambridge University Press, 1987.
- The mathematical theory of finite element methods, Susanne Brenner and Ridgway Scott, third edition, Springer, 2008.
Exams and homework: There will be homework sets and
computational projects, a midterm (on October 26, classtime), and a final exam
(December 18, 8:00AM-10:00AM).
Updated December 12, 2012