Math 272C (Numerical Partial Differential Equations)

Course Topics: Numerical Treatment of Partial Differential Equations
Instructor: Prof. Michael Holst (5161 AP&M, mholst@ucsd.edu; Regular Office Hours: TBA)
Term: Spring 2021
Lecture: 10:00a-10:50a MWF, Zoom (See Canvas for Zoom Links and info)
TA: None
Discussion: None

Main Class webpage: http://ccom.ucsd.edu/~mholst/teaching/ucsd/272c_s21/index.html

Canvas/Zoom:
    In 2020-2021 the course 272ABC will be based around the use of Canvas, which will be the place to find all materials for the course, including links to Zoom lectures and office hours, as well as any recorded lectures or materials that might be provided as part of the course. Note that this means that the webpage I created on my UCSD website for the course will not be updated after the first day of class; please use your Canvas account for the class going forward.
Textbook(s):
    I am listing three books for the course, all of which are free for students. The first book covers all of the topics of 272ABC at a survey level, in roughly the same order as the topics occur in 272ABC. The second book is an advanced book on the theory of finite element methods, and will be a reference for some of the more advanced material toward the end of each quarter. The third book is a reference for the tools from real and functional analysis, partial differential equations, and approximation theory that we will need throughout 272ABC.

  • Numerical Approximation of Partial Differential Equations.
    A. Quarteroni and A. Valli,
    Springer-Verlag, Berlin, 543 pages, 1994.
    This book is free to UCSD students and faculty, and a PDF version of the book can be found [ here ].

  • The Mathematical Theory of Finite Element Methods, Third Edition. S.C. Brenner and L.R. Scott,
    Springer-Verlag, Berlin, 397 pages, 2008.
    This book is free to UCSD students and faculty, and a PDF version of the book can be found [ here ].

  • Green's Functions and Boundary Value Problems.
    I. Stakgold and M. Holst,
    John Wiley & Sons, New York, NY, Third Edition, 855 pages, 2011.
    This book is free to UCSD students and faculty, and a PDF version of the book can be found [ here ].

    • NOTE: You will need to be coming from an official UCSD IP address, either from a campus machine or via a VPN to campus, in order to get free access to these three books. If you still have problems accessing the books, please contact me (email above).
Printable Syllabus: A printable version of this webpage can be found [ here ].


CATALOG DESCRIPTION:

272C. NUMERICAL PARTIAL DIFFERENTIAL EQUATIONS III (4)         [Spring 2021]
Time dependent (parabolic and hyperbolic) PDEs. Method of lines. Stiff systems of ODEs. Space-time finite element methods. Adaptive meshing algorithms. A posteriori error estimates. Prerequisites: MATH 272B or consent of instructor.

NOTE: For reference, here is the catalog description of the other two parts of 272 that will run in Fall 2020 and Winter 2021:

272A. NUMERICAL PARTIAL DIFFERENTIAL EQUATIONS I (4)         [Fall 2020]
Survey of discretization techniques for elliptic partial differential equations, including finite difference, finite element and finite volume methods. Lax-Milgram Theorem and LBB stability. A priori error estimates. Mixed methods. Convection-diffusion equations. Systems of elliptic PDEs. Prerequisites: graduate standing or consent of instructor.

272B. NUMERICAL PARTIAL DIFFERENTIAL EQUATIONS II (4)         [Winter 2021]
Survey of solution techniques for partial differential equations. Basic iterative methods. Preconditioned conjugate gradients. Multigrid methods. Hierarchical basis methods. Domain decomposition. Nonlinear PDEs. Sparse direct methods. Prerequisites: MATH 272A or consent of instructor.


GENERAL THEME OF THE COURSE: Math 272ABC was designed as a survey of numerical methods for partial differential equations, with a strong focus on the rigorous analysis of these methods. Both finite difference and finite volume methods can be interpreted as particular variations of Petrov-Galerkin methods, and as a result they have a much closer relationship to finite element methods than is generally known. We will cover most of the topics listed in the catalog description in the order listed, and we will also cover a few advanced topics. Each of the quarters will involve some review of basic tools in real and functional analysis and approximation theory, as needed for the particular topics of the quarter.


DEVIATION FROM THE CATALOG DESCRIPTION: We will stick mostly to the selection and the ordering of the topics listed in the catalog for each of the three quarters of 272ABC, but there will be some deviations from the catalog description, primarily some additional topics in all three quarters that are not listed in the abbreviated catalog descriptions. Most of the additional material in all three quarters will revolve around handling nonlinear problems (analysis, discretization, solvers), and in the Winter and Spring quarters there will be some updated material on the modern version of mixed methods (namely, the finite element exterior calculus).


GRADES AND IMPORTANT DATES: There will be 3-4 sets of homework problems passed out about every 2-3 weeks, which will be based on the materials covered in lecture. There will also be a single exam at the end of the course, based (very) heavily on the homework problem sets. Your grade in the course will be based on these three metrics:
  1. Homeworks Sets (50%)
  2. Final Exam (25%)
  3. Lecture Attendence (25%)
Important dates:
First lecture: MON 03/29
Last lecture: FRI 06/04
Memorial Day (no lecture): MON 05/31
Finals week: SAT-SAT, 06/05-06/11
Final Exam: TBA


WHERE TO FIND MORE DETAILED COURSE INFO (CANVAS) The webpage you are viewing is meant to be a very short point-of-contact for students looking for basic information about the course. More detailed information is found on your Canvas page for this course, which will play the role as the main information and materials hub for the course, including links and connection information for all of the Zoom lectures and other meetings.

ZOOM LECTURE ACCESS FOR REGISTERED STUDENTS: The Canvas website is now available and it contains the Zoom link for all of the lectures, beginning with the first lecture. The first time you connect to the Zoom lectures for this class you will need to fill out a simple registration form, but after than simply clicking on the link will take you to the lecture.

ZOOM LECTURE ACCESS FOR WAITLISTED AND/OR AUDITING STUDENTS: If you are a UCSD student and are either waitlisted, or want to audit the course, I can give you access to the Canvas page as an "observer", so you will have access to all materials for the course (Zoom lectures, lecture notes, etc). I just need to know your official UCSD email to register you.

NON-UCSD/NON-CANVAS ACCESS: If you are not officially registered in the course and so cannot access the course and Zoom links through Canvas, please contact me via email (address above) and I will try to get you access in some other way. Please put "272" somewhere in the subject line so that I can make sure to address it sooner rather than later.