Math 273B (Advanced Techniques in Computational Mathematics)

Course Topics: Computational Nonlinear PDE and Multiscale Modeling
Instructor: Prof. Michael Holst (5739 AP&M,
Term: Winter 2013
Lecture: 1:00p-1:50p MWF, APM 2402
Class webpage:
Textbook(s): I. Stakgold and M. Holst, Green's Functions and Boundary Value Problems.
                      John Wiley & Sons, New York, NY, Third Edition, 877 pages, 2011.

Brief review of tools from nonlinear functional analysis for numerical treatment of nonlinear PDE. Numerical continuation methods, pseudo-arclength continuation, gradient flow techniques, and other advanced techniques in computational nonlinear PDE. Project-oriented; projects designed around problems of current interest in science, mathematics, and engineering. Prerequisites: consent of instructor.

GRADES, HOMEWORKS, EXAMS, AND IMPORTANT DATES: Your grade in the course is based on your project and your 20-minute project presentation during the time of the final.
  • Final Exam/Project Time/Place: Tue 3/19/2013, 11:30a-2:29p, 2402 AP&M
    (NOTE: Moved up from Fri 3/22/2013; We may have to use a different room than 2402.)

SCHEDULE OF LECTURES: We will cover the four major topics listed below. The first topic will be briefly reviewed the first week, with some of the sub-topics inserted throughout the quarter when it makes sense. The other three major topics will each take about three weeks of the quarter. The individual sub-topics listed within each of the major topics will be covered in varying degree, depending on the available lecture time.
  1. Linear Analysis:
    Banach and Hilbert spaces. Linear operators, adjoints, four fundamental subspaces. Operator norms. Bounded, unbounded, closed, compact, projection operators. Fredholm alternative. Semi-groups. Linear functionals, dual space, reflexive and separable spaces. Strong, weak, weak-* topologies. Riesz, Hahn-Banach, Banach-Schauder, and Banach-Steinhaus Theorems.
  2. Elliptic and Parabolic Partial Differential Equations:
    Dirichlet and Neumann problems for elliptic and parabolic equations. Classical, strong, weak solutions. Distributions and fundamental solutions. Definition of Sobolev spaces. Ellipticity, inf-sup conditions, energy estimates, Lax-Milgram Theorem, existence, uniqueness. Maximum principles, a priori L-infinity estimates, Harnack inequality. Petrov-Galerkin methods, discrete inf-sup conditions, Babuska-Brezzi-Ladyzenskaya Theorem, a priori energy error estimates for abstract linear problems, applications to linear PDE.
  3. Nonlinear Analysis:
    Nonlinear maps in Banach spaces, variations, Gateaux and Frechet derivatives. Taylor expansion, Newton's Method. Inverse and Implicit Function Theorems, Bifurcation. Stationarity of functionals. Weakly closed sets, weak-lower-semi-continuity, reflexive Banach spaces, weak convergence. Existence of minimizers. Best approximation in Banach spaces, a priori energy error estimates for abstract nonlinear problems, applications to nonlinear PDE.
  4. Fixed-Point Techniques:
    Fixed-point iteration, rates of convergence. Banach Fixed-Point Theorem. Ordered Banach spaces, maximum principle, monotone increasing maps. Method of sub- and super-solutions, compatible barriers. Extension of the Brouwer Theorem to the Schauder Fixed-Point Theorem. Basic techniques of bifurcation theory, homotopy methods, and numerical continuation methods. Applications to nonlinear PDE.