Jean-Luc Thiffeault

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.

Textbooks

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):

Homework/Grading

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

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

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:
Office Hours:	Tue 10:00–11:00, Thu 12:30–13:15

Math 715: Methods of Computational Mathematics II: Spring 2018

Description

Textbooks

Homework/Grading

Homework Sets

Schedule of Topics