Jean-Luc Thiffeault's Homepage

Math 715: Methods of Computational Mathematics II: Spring 2018

Lecture Room: B123 Van Vleck Hall
Lecture Time: 14:30–15:45 TuTh
Lecturer: Jean-Luc Thiffeault
Office: 503 Van Vleck
Phone: (608)263-4089
Email: jeanluc@[domainname],
where [domainname] is math point wisc point edu
Office Hours: Tue 10:00–11:00, Thu 12:30–13:15


Topics: numerical linear algebra, finite element methods, boundary integral method.

We will begin by covering a number of issues in numerical linear algebra, such as matrix decomposition theorems, conditioning and stability in the numerical solution of linear systems, and iterative methods. With these tools in hand, we will proceed to discuss the finite element method, continuous and discontinuous Galerkin methods, multigrid methods, and error estimates. We will also cover boundary element and boundary integral techniques for the numerical solution of PDEs recast into integral form.

Incoming students should be comfortable with programming (in C++, Fortran, Python, or Matlab, or...), should have completed coursework in linear algebra, and should have familiarity with solving partial differential equations.


There is no official course textbook. You may be interested in visiting the following supplementary texts, which will be placed on reserve in the Math Library (Van Vleck B224):


The final grade will be determined by scores on homework assignments, which will be both analytical and computational in nature. Feel free to discuss the homework with each other, but you are required to code/write up your own solutions. Your work and results should be communicated clearly: points will be deducted for lack of clarity!

Homework Sets

Homework set #1 (Due 02/08): HW #1 [UWlogo.jpg]

Homework set #2 (Due 03/01): HW #2

Homework set #3 (Due 03/22): HW #3

Homework set #4 (Due 04/12 04/24): HW #4

Schedule of Topics

lecture date pages topic
1 01/23 T&B 3–24 Norms
2 01/25 T&B 25–36 The Singular Value Decomposition
3 01/30 T&B 41–61 Projectors; QR Factorization & Gram–Schmidt
4 02/01 T&B 69–85 Householder Triangularization; Least Squares
5 02/06 T&B 89–100 Condition Number; Floating Point Arithmetic
6 02/08 T&B 102–116 Accuracy and Stability
7 02/13 T&B 116–127; 147–150 Stability of Backsubstitution; LU Factorization
8 02/15 T&B 147–170 LU Factorization; Stability of LU
9 02/20 T&B 172–187 Cholesky Factorization; Eigenvalue Problems
10 02/22 T&B 187–209 Schur Factorization; Rayleigh Quotient Iteration [note]
11 02/27 T&B 211–215 QR Algorithm
12 03/01 T&B 215–233 QR Algorithm with Shifts; Other Algorithms
13 03/06 T&B 234–240 Computing the SVD; Iterative Methods
14 03/08 T&B 250–274 Arnoldi Iteration; GMRES
15 03/13 T&B 313–318 Preconditioning
16 03/15 T&B 293–301 Conjugate Gradient Method; Intro to Elliptic Problems
17 03/20 Johnson 14–22 Three Equivalent Problems; Finite Element Method with Piecewise Linear Functions
18 03/22 Johnson 26–37 FEM in 2D; Function Spaces
03/27,29 Spring Break
19 04/03 Johnson 40–55 Natural vs Essential Boundary Conditions; Lax–Milgram Theorem
20 04/05 Johnson 57–62 Some examples of applications of Lax–Milgram
21 04/10 Johnson 67–81 Finite-Element Spaces
22 04/12 Johnson 84–93 Error estimates
23 04/17 Discussion of homework
24 04/19 Finite-element program example
25 04/24 Johnson 146–159 FEM for parabolic PDEs: discontinuous Galerkin method
26 04/26 Johnson 167–196 FEM for hyperbolic PDEs