No. |
Date |
Topics |
Introduction |
1 |
9/5 |
Introduction and motivation for studying numerical analysis of PDE;
derivation of heat equation, how numerical PDE is used |
Finite difference methods for elliptic problems |
2 |
9/7 |
Elliptic boundary value problems, Poisson's equation, the Sleipner disaster;
finite difference methods, derivation of 3-point centered differences
and the 5-point Laplacian |
3 |
9/10 |
consistency of the 5-point Laplacian, bounds on consistency error; implementation of
the 5-point Laplacian |
4 |
9/12 |
performance of the 5-point Laplacian; the discrete maximum principle; nonsingularity |
5 |
9/14 |
stability and convergence for the 5-point Laplacian; the concept of stability;
the fundamental estimate relating error and consistency error |
6 |
9/17 |
the fundamental theorem: consistency plus stability imply convergence;
L2 norms and eigenvalue analysis |
7 |
9/19 |
L2 stability for the five point Laplacian |
8 |
9/21 |
the 5-point Laplacian on curved domains; the Shortley-Weller formula;
consistency error |
9 |
9/24 |
stability analysis in weighted norms, second order convergence of
the 5-point Laplacian on curved domains |
Linear algebraic solvers |
10 |
9/26 |
Introduction to solvers: direct methods (Gaussian elimination,
Cholesky decomposition, band solvers, operation counts); iterative methods and residual
correction |
11 |
9/28 |
Classical iterations as splitting methods and one-point iterations;
convergence and its relation to the spectrum of the iteration matrix |
12 |
10/1 |
Analysis of Richardson iteration |
13 |
10/3 |
Numerical study of Richardson, Jacobi, Gauss-Seidel, and SOR;
analysis of symmetrized iterations and convergence of Gauss-Seidel |
14 |
10/5 |
Line search methods, the method of steepest descents |
15 |
10/8 |
Introduction to the conjugate gradient method |
16 |
10/10 |
The conjugate gradient method, finite termination property, efficient implementation |
17 |
10/12 |
Rate of convergence for conjugate gradients |
18 |
10/15 |
Preconditioning; implementation and convergence of preconditioned conjugate
gradients |
19 |
10/17 |
Multigrid methods; smoothers, restriction and prolongation |
20 |
10/19 |
Implementation and performance of multigrid methods; V-cycle, W-cycle, full multigrid |
Finite element methods |
21 |
10/22 |
Introduction to finite element methods; weak formulations of a 2nd order BVP; the Sobolev space
H1; traces; Poincaré inequality |
22 |
10/24 |
Hilbert space framework; Galerkin's method; variational formulation; Rayleigh-Ritz method; stiffness matrix
and load vector |
23 |
10/26 |
Midterm exam |
24 |
10/29 |
P1 finite elements, basis elements, P1 finite element method for the Laplacian on a uniform grid
and its relation to finite differences |
25 |
11/2 |
essential and natural BCs; weak formulation of Dirichlet, Neumann, mixed, and Robin BVPs; shape functions, DOFs, unisolvence |
26 |
11/5 |
Lagrange finite spaces |
27 |
11/7 |
Introduction to FEniCS |
28 |
11/9 |
FEniCS continued; local stiffness matrix, finite element assembly |
29 |
11/12 |
Bilinear forms and linear operators on Hilbert space;
the Riesz Representation Theorem and the Lax-Milgram Lemma |
30 |
11/14 |
The inf-sup condition and the dense range condition;
quasioptimality; stability, consistency, and convergence of finite elements |
31 |
11/16 |
Introduction to finite element approximation theory;
Poincaré inequalities, averaged Taylor series |
32 |
11/19 |
The Bramble-Hilbert lemma, polynomial preserving operators |
33 |
11/21 |
Finite element approximation theory: scaling;
L2 error estimates for the interpolant |
34 |
11/26 |
Scaling in H1 and shape regularity;
error estimates for the finite element solution in H1;
the Aubin-Nitsche duality argument; L2 estimates |
35 |
11/28 |
The Clément interpolant |
36 |
11/30 |
Error analysis for the Clément interpolant |
37 |
12/3 |
Residual-based a posteriori error estimation |
38 |
12/5 |
Error indicators and adaptivity |
39 |
12/7 |
Finite element methods for nonlinear problems; Picard iteration |
40 |
12/10 |
Linearization and Newton's method; finite elements for the
minimal surface equation |
41 |
12/12 |
Discussion, review, preview of next semester, dots |