| No. | Date | Topics |
|---|---|---|
| 1 | 1/17 | Finite differences for the heat equation; analysis of the forward difference method; conditional stability |
| 2 | 1/19 | Backward differences for the heat equation; Crank-Nicolson |
| 3 | 1/22 | Fourier (von Neumann) analysis for the heat equation; the advection equation and finite differences for it; CFL condition |
| 4 | 1/24 | Fourier analysis for the advection equation; the Lax-Friedrichs method; introduction to finite elements for the heat equation |
| 5 | 1/26 | Convergence of semidiscrete FEM for the heat equation; convergence of the full discrete method with backward differences; the elliptic projection |
| 6 | 1/29 | The biharmonic and the thin plate equation; variational forms; boundary conditions |
| 7 | 1/31 | C1 elements; the Hermite quintic (Argyris) element; approximation theory by Bramble-Hilbert, dilation, and compactness |
| 8 | 2/2 | The Hsieh-Clough-Tocher composite element |
| 9 | 2/5 | Nonconforming finite elements for the Poisson equation; the Crouzeix-Raviart element; consistency error |
| 10 | 2/7 | Error analysis for the Crouzeix-Raviart element in H1 and L2 |
| 11 | 2/9 | Nonconforming H1 elements of higher degree and in 3D; the Morley element |
| 12 | 2/12 | Convergence theory for the Morley element |
| 13 | 2/14 | Mixed formulation of the Poisson equation; weak formulation, variational formulation, complementary energy and Lagrange multipliers |
| 14 | 2/16 | Mixed formulation: boundary conditions, variable coefficients, lower order term; numerical experiments for FE Galerkin methods |
| 15 | 2/19 | Numerical study of different finite element spaces for the mixed Poisson equation |
| 16 | 2/21 | Duality in Hilbert spaces |
| 17 | 2/23 | Closed Range theorem; Brezzi's theorem |
| 18 | 2/26 | Stability of mixed Galerkin methods; example: mixed finite elements in 1D |
| 19 | 2/28 | The lowest order Raviart-Thomas elements |
| 20 | 3/2 | Error estimates for Raviart-Thomas elements |
| 21 | 3/7 | Midterm |
| 22 | 3/9 | Higher order Raviart-Thomas elements |
| 23 | 3/19 | Duality estimates for the Raviart-Thomas elements; BDM elements |
| 24 | 3/21 | Finite elements for the Stokes equations; Fortin operators; stability of the P2-P0 Stokes element |
| 25 | 3/23 | Computational simulations of Stokes and Navier-Stokes flow |
| 26 | 3/26 | The mini element for Stokes; other stable Stokes elements in 2D and 3D |
| 27 | 3/28 | Introduction to Finite Element Exterior Calculus (FEEC) |
| 28 | 3/30 | Homological algebra: chain complexes, homology, the simplicial chain complex of a simplicial complex |
| 29 | 4/2 | Homological algebra: chain maps, cochain complexes, the simplicial cochain complex |
| 30 | 4/4 | The de Rham complex, de Rham's theorem |
| 31 | 4/6 | Unbounded operators in Hilbert space |
| 32 | 4/9 | Hilbert complexes, dual complex, harmonic forms, Hodge decomposition |
| 33 | 4/11 | Poincaré's inequality; the abstract Hodge Laplacian |
| 34 | 4/13 | Equivalence of formulations of Hodge Laplacian; well-posedness |
| 35 | 4/16 | The Hodge Laplacian on a 3D domain; boundary conditions |
| 36 | 4/20 | Galerkin methods for the Hodge Laplacian; problems with the primal formulation; subcomplex property and bounded cochain projections |
| 37 | 4/23 | Preservation of cohomology; the discrete Poincaré inequality; stability and convergence for the mixed Galerkin method |
| 38 | 4/25 | Exterior algebra |
| 39 | 4/26 | Exterior calculus: the exterior derivative |
| 40 | 4/27 | Integration of differential forms, Stokes theorem; the L2 theory of differential forms |
| 41 | 4/30 | Shape functions for finite element differential forms, the polynomial de Rham complex |
| 42 | 5/2 | The Koszul complex, the homotopy formula, trimmed polynomial spaces |
| 43 | 5/3 | Finite element differential forms: degrees of freedom and unisolvence; finite element de Rham subcomplexes; commutativity of the canonical projection |
| 44 | 5/4 | Mixed finite elements for the Hodge Laplacian; the Whitney forms and a proof of de Rham's theorm |