MATH 8446 Tentative Lecture Schedule

Finite elements for plates, C1 finite elements
No. Date Topics
1 1/18 Elasticity and elastic plates, variational, weak, and strong form of the clamped Kirchhoff plate model
2 1/20 Simply-supported plate, Babuška's paradox, Hermite elements in 1D
3 1/23 C1 finite elements; Hermite cubic and Hermite quintic (Argyris) elements; approximation properties of Argyris elements
4 1/25 Reduced Hermite cubic element (Bell's triangle); HCT element
Nonconforming finite elements
5 1/27 Nonconforming elements, P1 nonconforming elements, convergence theory
6 1/30 Consistency error bound for P1 nonconforming elements
7 2/1 The Morley nonconforming plate finite element
Mixed finite elements
8 2/3 Mixed finite elements for the Poisson equation
9 2/6 Boundary conditions in the mixed formulation, the Stokes equations, the structure of saddle point problems
10 2/8 Duality in Hilbert and Banach spaces, ranges and annihilators
11 2/10 Closed Range Theorem, Brezzi's theorem
12 2/13 Proof of Brezzi's theorem and well-posedness of saddle point problems
13 2/15 Stability of Galerkin methods for saddle point problems
14 2/17 The lowest order Raviart-Thomas space, P1-
15 2/20 The Raviart-Thomas projection and stability of the mixed method
16 2/22 Higher order Raviart-Thomas spaces (Pr-); unisolvences
17 2/24 Stability of higher order Raviart-Thomas spaces (Pr-); improved error estimates; the BDM family
The Stokes equations
18 2/27 Finite elements for the Stokes equations, some numerical results
19 2/29 Fortin operators; stability of the P2-P0 Stokes elements
20 3/2 The mini element and its analysis; other stable Stokes elements in 2- and 3-D
Elasticity
21 3/5 Introduction to elasticity: displacement, stress, strain, constitutive equations, elastic moduli, isotropy
3/7 no class
3/9 Midterm
22 3/19 Boundary value problems of linear elasticity, weak formulations
23 3/21 Korn's inequality, coercivity, Galerkin methods and estimates, numerical examples
24 3/23 The pure traction problem, rigid motions, compliance tensor, trace-free (deviatoric)/pure-trace decomposition, incompressibility and near incompressibility; the mixed (stress-displacement) formulational of elasticity
3/26 no class
25 3/28 The Johnson-Mercier elements for mixed elasticity
26 3/30 Airy stress function, compatibility of strain, the elasticity complex
27 4/2 Completion of unisolvence proof and stability analysis for Johnson-Mercier
4/4 no class
28 4/6 The AW mixed elements for elasticity, unisolvence
29 4/9 Mixed finite elements for nearly incompressible and incompressible elasticity, weak symmetry
Ten lectures on Finite Element Exterior Calculus
lectures will be 75 minutes
30 4/11 Introduction to finite element exterior calculus; basic homological algebra: chain complexes, chain maps, and homology; simplicial homology and de Rham cohomology
4/13 no class
31 4/16 Unbounded operators on Hilbert space, graph norm, closed operators, adjoints
32 4/18 Duality between null space and range, closed range theorem, examples with common function spaces
33 4/20 Hilbert complexes, dual complex, three key properties of closed Hilbert complexes: harmonic forms, Hodge decomposition, Poincaré inequality; the L2 de Rham complex
34 4/23 The abstract Hodge Laplacian, equivalence of strong, primal weak, and mixed weak formulations; well-posedness
35 4/25 Galerkin methods for the abstract Hodge Laplacian; three key properties: approximation, subcomplexes, bounded cochain projects; three major conclusions: preservation of cohomology, approximation of harmonic forms, uniform Poincaré inequality
36 4/27 Stability and convergence of Galerkin methods for the abstract Hodge Laplacian, basic error estimate, improved error estimates
37 4/30 Exterior algebra and calculus: algebraic forms, differential forms, the de Rham complex
38 5/2 The Koszul complex and the homotopy formula; the spaces Pr and Pr- spaces, unisolvence, finite element de Rham subcomplexes
39 5/4 Bounded cochain projections, applications