MATH 8445   Fall 2011 Lecture Schedule

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