Advanced Numerical Analysis: Finite Element Methods Coordinates
Spring 2006, Mondays 9:20 - 11:10 am
Office: WWH 712
Office Hours: Mondays 11:30am - 12:30pm or
drop by any time, or send email or call for an appointment.
Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics
by Dietrich Braess. Cambridge University Press. Second Edition.
Other Good Books
The Mathematical Theory of Finite Elements by Susanne Brenner and Ridgway
Scott. Second Edition. Springer.
Numerical Analysis of the Finite Element Method by Philippe Ciarlet.
University of Montreal Press.
Numerical Solution of Partial Differential Equations by the Finite
Element Method by Claes Johnson. Cambridge University Press.
- January 23. Variational formulation of Poisson's equation. Dirichlet
and Neumann boundary conditions. V-ellipticity. Bilinear form and linear
functionals representing the data. Triangulation of the region. Piece-wise
polynomial, continuous finite element spaces. Conforming finite elements
in H^1; continuity between elements required. Examples of finite elements
with increasing degrees of the polynomials. The interpolation problem and
its role in the construction of basis elements for the finite element
spaces. Cea's lemma.
- January 30. The biharmonic problem with Dirichlet boundary conditions.
The equations of linear elasticity. C^1 elements, which are conforming
in H^2. Barycentric coordinates; reference elements and affine mappings
to the elements of the triangulation. Sparsity of the stiffness matrix.
An example of an elliptic problem with a harmonic solution and smooth
boundary values which has a singularity at a point on the boundary.
An example of function in H^1 which is unbounded.
- February 6. Lipschitz regions. Poincare and Friedrichs inequalities
derived by elementary means and by using Rellish's theorem. Interpretation
in terms of the first and second eigenvalues of certain elliptic
problems with different boundary conditions. The use of
Poincare's inequality to bound the interpolation error on individual
elements and the use of these results to obtain bounds on the error of
the finite element solution. A trace theorem for H^1. Sobolev spaces
including those with fractional order. Sobolev's inequality, in particular
the order required to guarantee continuity everywhere.
- February 13. Aubin-Nitsche's bounds for the L_2-error of the
finite element solution; the role of regularity of the solution and
convexity of Lipschitz regions. An example of a nonconforming but
useful element due to Crouzeix and Raviart. Motivation. Variational
crimes, regions which are not unions of regular elements, numerical
quadrature and nonconforming elements. Strang's first and second lemmas.
- February 20. NYU holiday.
- February 27. Strang's second lemma. Error bound for non-conforming
piece-wise linear finite elements. Comments on Morley elements. How
to turn data on the coordinates of the elements into stiffness matrices;
the process of assembling element stiffness matrices. Data structures
for the stiffness matrices. Band matrices and a different format for
storing sparse matrices. Best possible bounds on the work and storage
required to solve the linear systems arising in finite element approximations.
- March 6. A mesh model for understanding fill-in when using Cholesky's
method. Nested dissection ordering. Isoparametric elements based on triangles
and quadrilaterals with one curved side. Incompressible stationary Stokes
equations and a mixed formulation of flow in porous media. Approximation
of saddle point problems with pairs of finite element spaces. Error bounds
for the first unknown using a Strang lemma. (Note that a good introduction
to mixed finite element is Chapter 12 of the second edition of the
Brenner Scott text book listed
above on this home page.)
- March 13. Spring recess.
- March 20. Solving div(u) = g in two dimensions and on a region with
reasonable boundary. The inf-sup condition and its relation to the previous
result. Description of Taylor-Hood elements. Almost incompressible elasticity
and mixed finite elements. Matrix analysis of the resulting saddle-point
problem providing bounds on the solution components in terms of the inf-sup
constant and the Lame parameters of linear elasticity. The inf-sup constant
as the smallest eigenvalue of a generalized eigenvalue problem with a
Schur complement and a mass matrix.
- March 27. A possibly new error bound for mixed finite element methods
relying of the stability result established lst week. Mixed finite element
formulation of almost incompressible elasticity. Verifying inf-sup stability
using an approach due to Rolf Stenberg.
- April 3. Introduction to iterative methods for system of
arising from finite element problems. Overlapping Schwarz methods. Block
Jacobi methods. An outline of the abstract Schwarz theory. Handout: Chapter
1 of the Toselli and Widlund monograph.
- April 10. More on the abstract Schwarz theory.
Two level overlapping Schwarz methods. Theory in the abstract frame work.
- April 17. Iterative substructuring methods for two subdomains, in
particular the Dirichlet-Neumann algorithms. A few words about alternative
algorithms; cf.\ chapter 1 of the Toselli and Widlund monograph. An algorithm
due to Bramble, Pasciak, and Schatz for many subdomains and two dimensions.
An outline of how we can estimate the C_0 parameter which gives a lower
bound on the eigenvalues of the preconditioned operator.
- April 24. Recent research on iterative substructuring methods,
in particular, dual-primal FETI methods and BDDC algorithms. The spectra
of dual-primal FETI and BDDC methods are almost the same for the same
set of primal constraints.
- May 1. No class. I will be celebrating my sister's birthday in Stockholm.
- May 8. Make-up class to replace the lecture of May 1. Review of
multi-grid algorithms in the perspective of the abstract Schwarz theory.
Basic ideas; smoothing with damped Jacobi and Gauss-Seidel. Recursive
definition of multigrid methods. Yserentant's hierarchical basis method
and an outline of its theory. Why it fails in three dimensions. Outline
of the theory for V-cycle multi-grid.