Math 552 Course Schedule

Content Info Schedule Homework Hints Q & A Programs Software		MATH 552 Numerical Solution of PDEs Spring 2013 Course Outline and Tentative schedule Many well-known and widely used mathematical models of interesting and important scientific and engineering problems are formulated as partial differential equations. These models, due to their enormous complexity and possible singularity in their solutions, are often challenging problems to be solved numerically. This course will address both the analytical and the computational aspects of the subject. While offering an overview to various popular methods and well established theory, the students will also get a hands-on experience in through the solution of some models problems by themselves. A tentative outline is as follows: Introduction to the course and review of PDEs (1 lecture) Basic methods for numerical PDEs: an illustration via a model Poisson Equation (5 lectures)                Finite difference method                Finite element method                Finite volume method                Spectral method                Approximation and convergence properties More on elliptic equations in general domain (5 weeks)                Fundamental theory, maximal principle,                Variational principle, regularity                Structured and unstructured meshing                Basic finite difference schemes, boundary treatments                Discrete maximal principle and M-matrices                Convergence and Error estimates                Finite element spaces and basic error estimates                Conforming finite element method,                More general finite element methods                Finite volume and co-volume methods                Problems in exterior and infinite domain                Artificial boundary conditions                Boundary integral and boundary element methods                Methods for solving the discrete systems                Domain decomposition and parallel algorithms                Nonlinear variational problems Parabolic problems (2.5 weeks)                Methods of lines                Finite difference methods, Stability analysis                Fully discrete schemes,                Alternating and splitting methods,                Explicit-Implicit schemes Hyperbolic/convection dominated problems (2.5 weeks)                Basic difference approximations,                Upwinding, Dissipation and Dispersion,                Stability analysis,                Methods for scalar 1d conservation laws, Shocks                Methods foe Hamilton-Jacobi equation Schrodinger equation (1 week)                Splitting methods Additional topics (1-2 weeks)                Fluids, Electromagnetics, Phase Transitions,                Optimal Control, Inverse Problems No. Date Topics 01 01/10 02 01/15