Math 273B (Advanced Techniques in Computational Mathematics II)

Instructor: Prof. Michael Holst (5739 AP&M, mholst@math.ucsd.edu)
Term: Winter 2010
Lecture: 9:30a-10:45a TuThu, APM 5402
Class webpage: http://ccom.ucsd.edu/~mholst/teaching/ucsd/273b_w10/

Specific Course Topics for Winter 2010:
  • As part of a book project with Ivar Stakgold, we will give a series of lectures in the Winter and Spring 2010 quarters on the theory and numerical treatment of nonlinear elliptic equations. The focus of the Winter 2010 quarter is on mathematical tools in Partial Differential Equations and Nonlinear Analysis, as needed for the numerical treatment of nonlinear elliptic equations.
Textbooks/References:
  • Krey78: E. Kreyszig. Introductory Functional Analysis with Applications. John Wiley & Sons, Inc., New York, NY, 1990.
  • StHo10: I. Stakgold and M. Holst. Green's Functions and Boundary Value Problems. John Wiley & Sons, Inc., NY, NY, third edition, 2010.
  • ReRo93: M. Renardy and R. C. Rogers. An Introduction to Partial Differential Equations. Springer-Verlag, New York, NY, 1993.
  • Evan98: L. C. Evans. Partial Differential Equations. American Mathematical Society, Providence, RI, 1998.
  • Zeid91a: E. Zeidler. Nonlinear Functional Analysis and its Applications I: Fixed Point Theorems. Springer, NY, NY, 1991.
  • FuKu80: S. Fucik and A. Kufner. Nonlinear Differential Equations. Elsevier Scientific Publishing Company, New York, NY, 1980.
  • Kesa89: S. Kesavan. Topics in Functional Analysis and Applications. John Wiley & Sons, Inc., New York, NY, 1989.

SYLLABUS: We will cover the five major topics listed below, roughly every two weeks of the quarter. The individual topics listed within each of the five major topics will be covered in varying degree, depending on the available lecture time.
  1. Linear Analysis (Krey78,StHo10):
    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. Hyperbolic Partial Differential Equations (ReRo93,Evan98):
    First order systems and characteristics. Classification and well-posedness. Cauchy problem for hyperbolic equations, wave equation, conservation laws. Weak solutions, jump conditions, shocks. Weak, strong, symmetric hyperbolicity. Local and global existence.
  3. Elliptic and Parabolic Partial Differential Equations (Evan98,StHo10):
    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.
  4. Nonlinear Analysis (Zeid91a,FuKu80,StHo10):
    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. Applications to nonlinear PDE.
  5. Fixed-Point Techniques (Zeid91a,Kesa89,StHo10):
    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. Homotopy and continuation methods. Applications to nonlinear PDE.