Lecture Room: | B211 Van Vleck Hall |

Lecture Time: | 12:05–12:55 MWF |

Lecturer: | Jean-Luc Thiffeault |

Office: | 503 Van Vleck |

Phone: | (608)263-4089 |

Email: | |

Office Hours: | Wed and Thu 11:00–11:45 |

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):

- Trefethen & Bau, Numerical Linear Algebra (SIAM)
- Golub & Van Loan, Matrix Computations (JHU Press)
- Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method (Dover)
- Brenner & Scott, The Mathematical Theory of Finite Element Methods (Springer)
- Briggs, Henson & McCormick, A Multigrid Tutorial (SIAM)
- Higham, Accuracy and Stability of Numerical Algorithms (SIAM)

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 2/17/2014): HW #1

Homework set #2 (Due 3/03/2014): HW #2

Homework set #3 (Due
3/~~24~~31/2014): HW #3

Homework set #4 (Due 4/21/2014): HW #4

Homework set #5 (Due 5/12/2014): HW #5

lecture |
date |
pages |
topic |

1 | 01/22 | T&B 3–24 | Norms |

2 | 01/24 | T&B 25–31 | The Singular Value Decomposition |

3 | 01/27 | T&B 32–47 | More on SVD; Projectors |

4 | 01/29 | T&B 48–62 | QR Factorization & Gram–Schmidt |

5 | 01/31 | T&B 69–76 | Householder Triangularization |

6 | 02/03 | T&B 69–92 | Least Squares; Condition Number |

7 | 02/05 | T&B 93–96 | Condition Number |

8 | 02/07 | T&B 97–107 | Floating Point Arithmetic |

9 | 02/10 | T&B 108–120 | Accuracy and Stability |

10 | 02/14 | T&B 121–128 | Stability of Backsubstitution; LU Decomposition |

11 | 02/17 | T&B 147–162 | LU Factorization |

12 | 02/19 | T&B 163–175 | Stability of LU; Cholesky Factorization |

13 | 02/21 | T&B 175–188 | Cholesky Decomposition; Eigenvalues |

14 | 02/24 | T&B 190–207 | Rayleigh Quotient; Inverse Iteration |

15 | 02/26 | T&B 207–215 | Rayleigh Quotient Iteration ; QR Algorithm |

16 | 02/28 | T&B 215–233 | QR Algorithm with Shifts; Other Algorithms |

17 | 03/03 | T&B 234–240 | Computing the SVD |

18 | 03/05 | – | Jacobi & Gauss–Seidel Iteration |

19 | 03/07 | T&B 250–265 | Arnoldi Iteration for Eigenvalues |

20 | 03/12 | – | Discussion of Homework Problem |

21 | 03/12 | T&B 266–275 | GMRES |

22 | 03/14 | T&B 293–298 | Conjugate Gradient Method |

23 | 03/24 | T&B 298–301 | Convergence of CG Method; Intro to Elliptic Problems |

24 | 03/26 | Johnson 14–18 | Three equivalent problems |

25 | 03/28 | Johnson 18–22, 26–29 | Finite Element Method with Piecewise Linear Functions; FEM in 2D |

26 | 03/31 | Johnson 29–35 | FEM in 2D (cont'd); Hilbert Spaces |

27 | 04/02 | Johnson 35,38,40–42 | Hilbert Spaces; Natural vs Essential Boundary Conditions |

28 | 04/04 | Johnson 50–57 | Lax–Milgram Theorem; Error Estimate |

29 | 04/07 | – | Hilldale Lecture at WID |

30 | 04/09 | Johnson 57–62 | Some examples of applications of Lax–Milgram |

31 | 04/11 | Johnson 67–81 | Finite-Element Spaces |

32 | 04/14 | Johnson 67–81 | More examples of FE Spaces |

33 | 04/16 | Johnson 84–92 | Error estimates |

34 | 04/18 | Johnson 84–92 | Error estimates (cont'd; proof) |

35 | 04/21 | Johnson 101–104 | Physical example: elasticity |

36 | 04/23 | Johnson 106–107 | Physical example: Stokes flow; Mixed methods |

37 | 04/25 | Johnson 146–159 | FEM for parabolic PDEs: discontinuous Galerkin method |

38 | 04/28 | Johnson 167–196 | FEM for hyperbolic PDEs |

39 | 04/30 | Johnson 214–223 | Boundary Integral Method |

40 | 05/02 | Johnson 224–230 | Numerical solution of Fredholm integral equations; Boundary Element Method |

41 | 05/05 | BHM 7–27 | Intro to Multigrid |

42 | 05/07 | BHM 31–43 | Intro to Multigrid (cont'd) |