## 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