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 and discrete
Green's functions.)
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: Construction of finite element
subspaces(Triangulation of a domain and barycentric coordinates.)
Lecture 7: Affine families of finite elements
(Properties of the mapping of the reference triangle to a general
triangle.)
Lecture 8: Error estimates for piecewise
linear interpolation
(Derivation of function and derivative error estimates using
multipoint Taylor series.)
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.)
Lecture 10: Order of Convergence and
other Finite Elements (H^1 and L^2 error estimates for Ritz-Galerkin
methods, rectangular and quadrilateral finite elements, C^1 finite elements.)
Lecture 11: Approximation of saddle point
problems (Abstract formulation of constrained minimization problems,
Lagrange multipliers, connection to saddle point problems, finite
element approximation.)
Lecture 12: Error estimates for the
approximation of saddle point problems (Brezzi's theorem and
error estimates using special interpolants.)
Lecture 13: Application to the mixed finite
element method for Poisson's equation (Lowest order Raviart-Thomas
element, construction of the Pi_h operator, and error estimates.)
Lecture 14: Application to the stationary Stokes
equations (Construction of the Pi_h operator for the P_2 -- P_0 Stokes
element, error estimates, and discussion of other stable elements.)
Lecture 15: Efficient solution of the linear
systems arising from finite element discretization (Optimization
methods: steepest descent, conjugate-gradient method.)
Lecture 16: Efficient solution of the linear
systems arising from finite element discretization (Multigrid.)
Lecture 17: 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 18: Finite difference methods
for the transport equation and the wave equation (Introduction of
some basic methods, domain of dependence, CFL condition.)
Lecture 19: Stability of difference
schemes for pure IVP with periodic intial data (Development of
algebraic criteria for stability, amplification matrices,
von Neumann stability condition.)
Lecture 20: Stability of difference
schemes -- examples (Applications of the abstract conditions
for stability)
Lecture 21: Qualitative properties of finite
difference schemes (Dissipation and dispersion of finite difference
schemes.)
Lecture 22: Finite element methods for
parabolic problems (Formulation and analysis of continuous time
Galerkin methods and fully discrete schemes.)
Lecture 23: A finite element method for
the transport equation (Formulation and analysis of the discontinuous
Galerkin method for the transport equation.)
Lecture 24: Approximation of hyperbolic
conservation laws (Method of characteristics, numerical flux
and discrete conservation, Godunov scheme.)