MATH 8445-8446
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 time-dependent problems.


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.

Other references: Similar material is covered in numerous texts. The following are all available online through the University of Minnesota libraries.


Exams: 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.

Additional resources:

Class expectations: 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