## 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.)