No. |
Date |
Topics |
Introduction |
1 |
9/7 |
Introduction and motivation for studying numerical analysis of PDE;
derivation of heat equation and Poisson equation, how numerical PDE is used,
the Sleipner disaster |
Finite difference methods for elliptic problems |
2 |
9/9 |
Difference methods, the 5-point Laplacian (derivation, implementation, experimentation) |
3 |
9/12 |
L∞–analysis of the 5-point Laplacian via the maximum
principle |
4 |
9/14 |
Norms and inner products, errors, consistency error, stability constant |
5 |
9/16 |
An abstract convergence framework; consistency and stability imply convergence; L2–stability of the 5-point Laplacian via discrete Fourier analysis (started) |
6 |
9/19 |
L2–stability of the 5-point Laplacian via discrete Fourier analysis (concluded) |
7 |
9/21 |
Dealing with a curved boundary |
8 |
9/23 |
Optimal analysis of the 5-point Laplacian on curved domains |
Linear algebraic solvers |
9 |
9/26 |
Introduction to solvers, direct methods, band solvers, residual
correction, classical iterations |
10 |
9/28 |
Classical iterations as splitting methods and one-point iterations; numerical performance |
11 |
9/30 |
Analysis of one-point iterations; application to Richardson iteration |
12 |
10/3 |
Analysis of Gauss-Seidel iteration |
13 |
10/5 |
Optimization, line search methods, the method of steepest descents |
14 |
10/7 |
The conjugate gradient method |
15 |
10/10 |
Rate of convergence of conjugate gradients |
16 |
10/12 |
Preconditioned conjugate gradients |
17 |
10/14 |
Multigrid |
18 |
10/17 |
Implementation and performance of multigrid methods; V-cycle, W-cycle, full multigrid |
Finite element methods |
19 |
10/19 |
Introduction to finite element methods; weak and variational formulation of 2nd order BVPs |
20 |
10/21 |
The Sobolev space H1, trace theorem, Poincaré inequality; essential and natural BCs; weak formulation of Dirichlet, Neumann, mixed, and Robin BVPs |
21 |
10/24 |
Galerkin's method, P1 finite elements |
22 |
10/26 |
General finite element spaces, shape functions, DOFs, unisolvence, Lagrange finite elements |
23 |
10/28 |
Review of various topics, finite element datastructures and assembly process |
24 |
10/31 |
Midterm exam |
25 |
11/2 |
Coercivity, inf-sup condition, well-posedness of weak problems |
26 |
11/7 |
Stability, consistency, and convergence of finite elements |
27 |
11/9 |
Introduction to FEniCS |
28 |
11/11 |
FEniCS continued, Poincaré inequalities |
29 |
11/14 |
Bramble Hilbert Lemma |
30 |
11/16 |
Finite element approximation theory and scaling |
31 |
11/18 |
Finite element approximation theory, continued |
32 |
11/21 |
L2 estimates, the Aubin-Nitsche duality argument |
33 |
11/23 |
The Clément interpolant |
34 |
11/28 |
Residual-based a posteriori error estimation |
35 |
11/30 |
Error indicators and adaptivity |
36 |
12/2 |
Finite element methods for nonlinear problems; Picard iteration; Newton's method; the
minimal surface equation |
Time-dependent problems |
37 |
12/5 |
Time-dependent problems, finite differences for the heat equation, semidiscretization, forward-centered discretization |
38 |
12/7 |
Fourier analysis, backward-centered and Crank-Nicolson schemes |
39 |
12/9 |
Finite element methods for the heat equation, semidiscretization |
40 |
12/12 |
Fully discrete finite element method for the heat equation |
41 |
12/14 |
Numerical examples, etc. |