| 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. |