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 |