Math 575 Lecture Notes: Spring, 2015

  • 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.) Revised 2/19/2015
  • Lecture 6: Definition and construction of finite element subspaces (Triangulation of a domain, shape functions, degrees of freedom, and barycentric coordinates.) Revised 2/17/2015
  • 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.) Revised 2/26/2015
  • 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.) Revised 3/24/2015
  • Lecture 10: A posteriori error estimates (Derivation and a posteriori error estimates and application to adaptive finite element methods.) Revised 3/9/2015, Revised 3/10/2015, Revised 3/24/2015
  • Lecture 11: Approximation of elliptic variational inequalities (Formulation and abstract approximation; application to the obstacle problem.) Revised 3/26/2015
  • Lecture 12: Efficient solution of the linear systems arising from finite element discretization (Optimization methods: steepest descent, conjugate-gradient method.) Revised 3/31/2015, Revised 4/2/2015
  • Lecture 13: Efficient solution of the linear systems arising from finite element discretization (Multigrid.) Revised 4/15/2015
  • 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.) Revised 4/15/2015
  • Lecture 16: Finite difference methods for the transport equation and the wave equation (Introduction of some basic methods, domain of dependence, CFL condition.) Revised 4/28/2015, Revised 5/6/2015
  • 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). Revised 4/23/2015 Revised 4/30/2015
  • Lecture 19: Finite element methods for parabolic problems (Formulation and analysis of continuous time Galerkin methods and fully discrete schemes.) Revised 4/28/2015 Revised 4/30/2015
  • Lecture 20: Approximation of parabolic variational inequalities. Revised 4/30/2015
  • Lecture 21: Space-time finite element methods for parabolic problems. Revised 4/30/2015