##
Math 575 Lecture Notes

Lecture 1: Finite Difference Methods
for Elliptic Problems
(Approximation of the Dirichlet problem for Poisson's equation;
discrete maximum principle.)
Lecture 2: Stability and Error Estimates
(Stability and error estimates for finite difference schemes for
Poisson's equation using the discrete maximum principle.)
**Revised 2/1/2011**
Lecture 3: Extensions of the Method
(Domains with curved boundaries, Neumann boundary conditions, higher
order approximations.)
Lecture 4: Finite Element Method
for Elliptic Equations - Introduction
(Preliminaries and variational formulations.)
Lecture 5: Finite Element Method
for Elliptic Equation
(Formulation as a minimization problem, Ritz-Galerkin approximation schemes,
basic error analysis.)
Lecture 6: Definition and construction
of finite element subspaces(Triangulation of a domain, shape functions,
degrees of freedom, and barycentric coordinates.) **Revised 2/22/2011**
Lecture 7: Families of finite elements
(Affine families and properties of the mapping of the reference triangle
to a general triangle, tensor product finite elements, C^1 finite elements.)
**Revised 2/23/2011**
Lecture 8: Error estimates for piecewise
linear interpolation
(Derivation of function and derivative error estimates for piecewise
linear approximation.) **Revised 2/23/2011**
Lecture 9: Error estimates by scaling
(Bramble-Hilbert lemma, effect of change of variable from the reference
triangle to an arbitrary triangle, interpolation error estimates for
piecewise polynomial approximation, application to Ritz-Galerkin approximation
schemes.) **Revised 2/23/2011**
Lecture 10: A posteriori error estimates
(Derivation and a posteriori error estimates and application to adaptive
finite element methods.)
Lecture 11: Approximation of elliptic
variational inequalities
(Formulation and abstract approximation; application to the obstacle problem.)
Lecture 12: Efficient solution of the linear
systems arising from finite element discretization (Optimization
methods: steepest descent, conjugate-gradient method.)
Lecture 13: Efficient solution of the linear
systems arising from finite element discretization (Multigrid.)
Lecture 14: Finite difference methods
for the heat equation (Introduction of some basic methods: forward
and backward Euler, Crank-Nicholson, proof of stability and error estimates.)
Lecture 15: Finite difference methods
for the transport equation and the wave equation (Introduction of
some basic methods, domain of dependence, CFL condition.)
Lecture 16: Stability of difference
schemes for pure IVP with periodic intial data (Development of
algebraic criteria for stability, amplification matrices,
von Neumann stability condition.)
Lecture 17: Stability of difference
schemes -- examples (Applications of the abstract conditions
for stability)
Lecture 18: Finite element methods for
parabolic problems (Formulation and analysis of continuous time
Galerkin methods and fully discrete schemes.)