
Math 273B (Advanced Techniques in Computational Mathematics)
Course Topics: Computational Nonlinear PDE and Multiscale Modeling
Instructor: Prof. Michael Holst
(5739 AP&M, mholst@math.ucsd.edu)
Term: Winter 2013
Lecture: 1:00p1:50p MWF, APM 2402
Class webpage:
http://ccom.ucsd.edu/~mholst/teaching/ucsd/273b_w13
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.
CATALOG DESCRIPTION:
273B. ADVANCED TECHNIQUES IN COMPUTATIONAL MATHEMATICS (4)
Brief review of tools from nonlinear functional analysis for numerical
treatment of nonlinear PDE.
Numerical continuation methods, pseudoarclength continuation, gradient flow
techniques, and other advanced techniques in computational nonlinear PDE.
Projectoriented; 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
20minute project presentation during the time of the final.
 Final Exam/Project Time/Place: Tue 3/19/2013, 11:30a2:29p, 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 subtopics inserted throughout the quarter when it makes sense.
The other three major topics will each take about three weeks of the quarter.
The individual subtopics listed within each of the major topics will
be covered in varying degree, depending on the available lecture time.
 Linear Analysis:
Banach and Hilbert spaces.
Linear operators, adjoints, four fundamental subspaces.
Operator norms.
Bounded, unbounded, closed, compact, projection operators.
Fredholm alternative.
Semigroups.
Linear functionals, dual space, reflexive and separable spaces.
Strong, weak, weak* topologies.
Riesz, HahnBanach, BanachSchauder, and BanachSteinhaus Theorems.
 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, infsup conditions, energy estimates,
LaxMilgram Theorem, existence, uniqueness.
Maximum principles, a priori Linfinity estimates,
Harnack inequality.
PetrovGalerkin methods,
discrete infsup conditions,
BabuskaBrezziLadyzenskaya Theorem,
a priori energy error estimates for abstract linear problems,
applications to linear PDE.
 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, weaklowersemicontinuity,
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.
 FixedPoint Techniques:
Fixedpoint iteration, rates of convergence.
Banach FixedPoint Theorem.
Ordered Banach spaces, maximum principle,
monotone increasing maps.
Method of sub and supersolutions, compatible barriers.
Extension of the Brouwer Theorem to the Schauder FixedPoint Theorem.
Basic techniques of bifurcation theory,
homotopy methods, and numerical continuation methods.
Applications to nonlinear PDE.
