##
Math 575 Lecture Notes: Spring, 2017

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.)
Lecture 3: Extensions of the Method
(Domains with curved boundaries, Neumann boundary conditions, higher
order approximations, more general elliptic operators.)
Lecture 4: Finite Element Method
for Elliptic Equations - Introduction
(Preliminaries and variational formulations.)
Lecture 5: Finite Element Method
for Elliptic Equations
(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.)
Lecture 7: Global bases and affine families
(Global bases for piecewise polynomial spaces; Affine families and
properties of the mapping of the reference triangle
to a general triangle.)
Lecture 8: Other families of finite elements;
error estimates for piecewise polynomial interpolation in 1-D
(Tensor product and quadrilateral finite elements, C^1 finite elements,
derivation of function and derivative error estimates for piecewise
linear approximation, generalization to higher order.)
Lecture 9: Error estimates in higher dimensions,
application to Ritz-Galerkin approximation schemes
(Interpolation estimates for piecewise polynomial approximation in 2-D,
error estimates for Ritz-Galerkin approximation schemes.)
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.)
**Revised February 28, 2017**
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.)
**Revised March 23, 2017**
Lecture 14: Iterative methods for
variational inequalities
Lecture 15: 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 16: Finite difference methods
for the transport equation and the wave equation (Introduction of
some basic methods, domain of dependence, CFL condition.)
Lecture 17: Stability of difference
schemes for pure IVP with periodic intial data (Development of
algebraic criteria for stability, amplification matrices,
von Neumann stability condition.)
Lecture 18: Stability of difference
schemes -- examples (Applications of the abstract conditions
for stability).
Lecture 19: Finite element methods for
parabolic problems (Formulation and analysis of continuous time
Galerkin methods and fully discrete schemes.)
Lecture 20: Approximation of parabolic
variational inequalities. **Revised April 18, 2017**
Lecture 21: Space-time finite element methods
for parabolic problems.
Lecture 22: A finite element method for the
transport problem. **Revised April 25, 2017**
Lecture 23: The finite volume method for
elliptic problems.
Lecture 24: Qualitative properties of finite
difference schemes (Dissipation and dispersion of finite difference
schemes.) **Revised April 27, 2017**