| No. | Date | Topics |
|---|---|---|
| Introduction | ||
| 1 | 9/6 | Introduction and motivation for studying numerical analysis of PDE |
| 2 | 9/8 | Elliptic boundary value problems, Poisson's equation |
| Finite difference methods for elliptic problems | ||
| 3 | 9/11 | derivation of 3-point centered difference |
| 4 | 9/13 | implementation and performance of the 5-point Laplacian; the discrete maximum principle; nonsingularity |
| 5 | 9/15 | stability and convergence for the 5-point Laplacian |
| 6 | 9/18 | eigenanalysis and L2 stability for the five point Laplacian |
| 7 | 9/20 | curved domains; the Shortley-Weller formula; stability analysis in weighted norms and second order convergence on curved domains |
| Linear algebraic solvers | ||
| 8 | 9/22 | Introduction to solvers, operation counts for direct solvers; iterative methods, residual correction, splitting methods, one-point iterations |
| 9 | 9/25 | Linear convergence, convergence of one-point iterations |
| 10 | 9/27 | Symmetrized iterations, convergence of Gauss-Seidel |
| 11 | 9/29 | Line search methods; method of steepest descents |
| 12 | 10/2 | The conjugate gradient method |
| 13 | 10/4 | Implementation and performance of conjugate gradients |
| 14 | 10/6 | Rate of convergence of conjugate gradients |
| 15 | 10/9 | Preconditioned conjugate gradients, incomplete Cholesky preconditioner |
| 16 | 10/11 | Multigrid methods; smoothers, restriction and prolongation |
| 17 | 10/13 | Implementation and performance of multigrid methods; V-cycle, W-cycle; multigrid as a preconditioner |
| Finite element methods | ||
| 18 | 10/16 | Weak solutions, Sobolev spaces, traces |
| 19 | 10/18 | Weak formulation of boundary value problems, natural |
| 20 | 10/20 | midterm exam |
| 21 | 10/23 | Galerkin's method, Rayleigh-Ritz method, stiffness matrix, load vector, P1 finite element method for the Laplacian on a uniform grid |
| 22 | 10/25 | Finite element spaces, shape functions and degrees of freedom, unisolvence |
| 23 | 10/27 | Lagrange finite spaces; finite element assembly |
| 24 | 10/30 | Introduction to FEniCS; a first program; Expressions and Functions |
| 25 | 11/1 | FEniCS continued; meshes, finite element spaces, forms, functionals, assembly, solve |
| 26 | 11/3 | Boundary conditions in FEniCS; applications |
| 27 | 11/6 | Bilinear forms and linear operators on Hilbert space; coercivity and the Lax-Milgram lemma |
| 28 | 11/8 | The inf-sup condition and the dense range condition; quasioptimality; stability, consistency, and convergence of finite elements |
| 29 | 11/10 | Introduction to finite element approximation theory; Poincaré inequalities, averaged Taylor series |
| 30 | 11/13 | The Bramble-Hilbert lemma, polynomial preserving operators |
| 31 | 11/15 | Finite element approximation theory: scaling; L2 error estimates for the interpolant |
| 32 | 11/17 | Scaling in H1 and shape regularity; error estimates for the finite element solution in H1 |
| 33 | 11/20 | The Aubin-Nitsche duality argument; L2 estimates |
| 34 | 11/22 | The Clément interpolant |
| 35 | 11/27 | Error estimation for the Clément interpolant; residuals and errors |
| 36 | 11/29 | A posteriori error estimation |
| 37 | 12/1 | Error indicators and adaptivity |
| 38 | 12/4 | Finite element methods for nonlinear problems; Picard iteration |
| 39 | 12/6 | Implementation and performance of Picard iterations |
| 40 | 12/8 | Linearization and Newton's method; finite elements for the minimal surface equation |
| 41 | 12/11 | error estimates for minimal surface equation |
| 42 | 12/13 | Newton's method for the p-Laplacian; continuation |