Numerical Analysis of Differential Equations
|Fall 2014, MWF 11:15-12:05 Vincent Hall 113|
|Spring 2014, MWF 11:15-12:05 Vincent Hall 207|
|Instructor: Douglas N. Arnold|
|Contact info: 512 Vincent Hall, tel: 6-9137, email:
About the course: This is a two-semester graduate level
introduction to the numerical solution of partial differential equations.
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 remainder of the course will
be devoted to finite element methods for elliptic problems,
and their analysis. In the first semester we will introduce finite element methods
for elliptic problems and analyze their behavior mathematicially. At the end of the
semester and during the second semester we will treat more advanced topics
in finite elements: time-dependent problems, C1 finite elements, nonconforming finite finite elements, mixed methods, and
various applications, such as the Stokes and Navier-Stokes equations, elasticity, and Maxwell's equations.
The course will include
computational examples and projects using
python, 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 numerous texts. The following are all available
online through the University of Minnesota libraries.
- 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, Society for Industrial and Applied Mathematics, 2002.
- The mathematical theory of finite element methods, Susanne Brenner and Ridgway Scott, third edition, Springer, 2008.
Final exam is Saturday May 16 from 1:30-3:30.
Here are some study problems.
The study problems for the midterm
are also relevant for the final.
Students are expected to attend the lectures and read the notes in coordination with them.
The lectures and notes will be terse at times, and students are expected to
work out and fill in the details. Students should become familiar
with Python including the NumPy package
and be able to run it in their preferred computing environment.
(For this, see the first five chapters of Langtangen's Primer
or any of the many tutorials on the web.) There will be homework assignments
both theoretical and computational, and a midterm and final exam.
Updated January 20, 2015