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 |
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 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
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 |