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;
L^{2} norms and eigenvalue analysis |

7 | 9/19 | L^{2} 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
H^{1}; 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 | P_{1} finite elements, basis elements, P_{1} 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;
L^{2} error estimates for the interpolant |

34 | 11/26 | Scaling in H^{1} and shape regularity;
error estimates for the finite element solution in H^{1};
the Aubin-Nitsche duality argument; L^{2} 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 |