| 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 | |