Advanced Numerical Analysis: Finite Element Methods
Advanced Numerical Analysis: Finite Element Methods
G22.2945, G63.2040
Fall 2001, Wednesdays 5:10-7:00pm, WWH 813
Instructor: Olof
B. Widlund
Coordinates
Office: WWH 712
Telephone: 998-3110
Office Hours: Drop by any time, or send email or call for appointment
Email: widlund@cs.nyu.edu
Homework
Homework 1 Assigned September 19,
due October 3.
Homework 2 Assigned September 26,
due October 10.
Homework 3 Assigned October 10,
due October 24.
Homework 4 Assigned October 31,
due November 14.
Homework 5 Assigned November 14,
due November 28 or at your earliest convenience.
Lectures
- September 5. A finite element method for a one-dimensional problem.
The difference between Dirichlet and Neumann boundary conditions. Green's
formula and a variational formulation of Poisson's equation. Examples
of continuous, piecewise polynomial finite element methods.
- September 12. NYU closed. A make-up
class will, if possible, be scheduled at the end of the semester.
- September 19. H^1-conforming finite elements must be continuous.
An example of a harmonic function with nice boundary values and a
singularity at a reentrant corner; the region is non convex and does
not have a smooth boundary. Friedrichs' and Poincar\'{e}'s inequalities.
Bounds from below of the first and second eigenvalues of the Laplacian
with Dirichlet (or mixed) and Neumann boundary conditions, respectively.
Lipschitz regions and a trace theorem for H^1 on such regions.
- September 27. Rellish's theorem and generalized Poincar\'{e}'s inequalities.
Approximation theory for finite element spaces. Examples of Lagrange finite
elements, Hermite finite elements on triangles. Finite element spaces on
quadrilaterals. C^1 finite elements due to Agryris, Bell, and Hsieh, Clough,
and Tocher.
- October 3. The Nitsche-Aubin error estimate. Regularity of elliptic
problems on regions with smooth boundaries or on convex regions. An
example of problem on a non-convex region with limited regularity for
smooth data. Error bounds for functions with limited smoothness
by quasi-interpolants. Aspects of implementation of finite element
methods.
- October 10. Why nonconforming finite element problems? Strang's Lemmas.
Analysis of nonconforming P_1 elements. Why such elements? Stokes' equations.
Saddle point problems.
- October 17. Morley's element for the biharmonic problem. L_2-bounds for
the error of nonconforming finite elements. Applications to the non-conforming
P_1 elements. Methods for regions with curved boundaries. Gauss-Lobatto
quadrature. Isoparametric elements. Dual norms and the inf-sup condition.
- October 24. inf-sup stability for the continuous Stokes problem. Mixed
finite element methods for Darcy's law. Norm of the solution operator for
saddle point problems expressed in terms of the inf-sup parameter of the
discrete mixed method. Error bounds for mixed methods in terms of the inf-sup
parameter and best approximation results.
- October 31. Saddle point problems with a penalty term with an example
from almost incompressible elasticity. An example where the bilinear form
a(.,.) is only elliptic on the subspace for which the constraints are
satisfied; cf. the minimal requirements in Brezzi's theorem on p. 132.
Fortin interpolation. Examples of inf-sup stable and inf-sup unstable methods
for Stokes equations. Tools for proving inf-sup stability of methods for
Stokes equations. The stability of the lowest order Taylor-Hood elements
following a paper by Rolf Stenberg.
- November 7. Examples of inf-sup stable finite element methods for
Stokes equations. Proof to inf-sup stability for the mini element.
Non-conforming
P_1 elements and a divergence free basis for two dimensions. Introduction to
domain decomposition: transmission conditions, the Neumann-Dirichlet algorithm.
- November 14. Schur complements, trace and extension theorems for finite
elments. Convergence rates of the Neumann-Dirichlet algorithm. The Neumann-
Neumann algorithm. The case of many subdomains. Introduction of the Schwarz
alternating method.
- November 21. Convergence of the two subdomain multiplicative Schwarz
method using arguments about Schur complements. The case of three and more
subdomains. The block-Jacobi preconditioner. Using overlap to enhance the
convergence. Additive Schwarz methods. Introduction to a theory for Schwarz
methods based on three assumptions.
- November 28. Abstract theory for Schwarz methods. Yserentant's
hierarchical basis method.
- December 5. More about Yserentant's method. Multigrid and BPX methods.
Two-level overlapping Schwarz methods.
Required Text
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. 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.
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