Course Information

Course Booklet Information Page

The course meets in G-block, from 1:30-2:30 PM on Mondays, Wednesdays, and Fridays, in Room BP-3. My office hours are Mondays and Wednesdays from 10:30 - noon, in Room BP-212.

There will be roughly 6 homework assignments over the semester, which will be due between 1 and 2 weeks after they are distributed. These assignments will combine some pencil-and-paper exercises with programming questions. Late homework will be penalized by 10% per day late - the only exception to this is that you are permitted one "freebie" in this system, that can be used to excuse a single day's lateness on a single assignment. You are permitted (and, generally, expected) to discuss the assignments with others in the class, but must hand in your own work individually.
The programming for this course can be done in any language that you choose. Matlab is probably the easiest to pick up if you are not familiar with any other. All Tufts students have access to Matlab in the ITS Computing Center @ Eaton Hall. A free alternative to Matlab is Octave. Other possibilities include using python with numpy/scipy/matplotlib. If you would like to use another option, please discuss this with me.
The first homework assignment will not be distributed until the third week of class. I strongly suggest that you use this time to familiarize yourself with matlab (or your language of choice) if you do not already know it. Many good tutorials are available online. There will be no leniency granted for late assignments due to unfamiliarity with programming.
Final Projects
The course has a required final project, with presentation. You may choose a topic related to anything discussed in the course (or not discussed in the course but related to numerical methods for PDEs) and investigate it in more depth. Topic choices must be approved by me before you start work on the project. You must produce a final paper, describing what you did in your project, as well as a 10-minute presentation to be done during the last week of class. Group projects are allowed, although the expectations for a group project increase in proportion to the size of the group.
Final Grade
Your final grade in the course will be computed as a weighted average of your homework and final project grades. In this calculation, the final project will count as 3 homework assignments, while your lowest homework assignment will count for one-half its normal value.

Academic Integrity
While there are no exams in this course, students are required to abide by the university guidelines on academic integrity. For full details, see this link.

Disability Services
If you are requesting an accommodation due to a documented disability, you must register with the Disability Services Office at the beginning of the semester. To do so, call the Student Services Desk at 617-627-2000 to arrange an appointment with Linda Sullivan, Program Director of Disability Services.

Learning Objectives
This course addresses the following learning objectives of the Ph.D. Program in Mathematics

  • 1.b. Clear understanding of key hypotheses and conclusions
  • 1.c. Synthesis of formal theory into a comprehensive picture of mathematical phenomena
  • 1.d. Application of general theory to specific examples
  • 1.e. Sharpening of intuition through appropriate counterexamples
  • 2.b. Efficient and/or coherent presentation of arguments in the form of proofs and/or data
  • 3.a. Explanation of key ideas and general strategies
  • 3.b. Motivation of underlying issues
  • 3.c. Clear oral presentation of arguments
  • 3.f. Thinking on one's feet; fielding questions
  • 5.b. Developing intuition about a new problem or situation
  • 5.d. Understanding the role of examples
  • 5.f. Computer literacy
  • Reference Books

    There are many books and websites that present material relevant to this course. The following are a few suggestions, but these are neither required reading nor a complete list of good references. Most of these are readily available electronically through the Tufts library.

  • Finite Difference Schemes and Partial Differential Equations, by John C. Strikwerda, SIAM 2004
  • Finite Difference Methods for Ordinary and Partial Differential Equations, by Randall J. Leveque, SIAM 2007
  • Principles of Computational Fluid Dynamics, by Pieter Wesseling, Springer 2001
  • A First Course in the Numerical Analysis of Differential Equations, by Arieh Iserles, Cambridge University Press 1996
  • Introduction to Computational PDEs, by Hans De Sterck and Paul Ullrich, University of Waterloo 2009
  • The Mathematical Theory of Finite Element Methods, by Suzanne Brenner and Ridgeway Scott, Springer 1994
  • Finite Elements: Theory, fast solvers, and applications in solid mechanics, by Dietrich Braess, Cambridge University Press 2001
  • Finite Elements and Fast Iterative Solvers with applications in incompressible fluid dynamics, by Howard Elman, David Silvester, and Andy Wathen, Oxford Science Publications 2005
  • Matrix Iterative Analysis, by Richard Varga, Springer 2000
  • Iterative Methods for Sparse Linear Systems, by Yousef Saad, SIAM 2003. Available here
  • Iterative Methods for Solving Linear Systems, by Anne Greenbaum, SIAM 1997
  • A Multigrid Tutorial, by William Briggs, Van Emden Henson, and Steve McCormick, SIAM 2000. Accompanying slides available here
  • Multigrid, by Ulrich Trottenberg, Cornelis Oosterlee, and Anton Schüller, Elsevier 2001
  • Schedule

  • 1/16: Introduction
  • 1/18: Review: Integration by Parts & Chain Rule
  • 1/23: Review: Taylor's Theorem
  • 1/25: One-way Wave Equation (Scott away - class with James)
  • 1/28: G+ block Consistency and Convergence, Stability
  • 1/30: G+ block Systematic von Neumann Analysis (HW1 distributed)
  • 2/1: Lax Convergence and Equivalence Theorems
  • 2/4: G+ block Operator Norms, Discrete-Time Fourier Transforms
  • 2/6: G+ block von Neumann Stability Analysis, Dissipation and Dispersion
  • 2/11: Numerical Dissipation and Dispersion (HW2 distributed)
  • 2/13: Implicit and Explicit Schemes
  • 2/15: Parabolic PDEs
  • 2/20: Elliptic PDEs and Weak Forms
  • 2/21: Ritz-Galerkin Approximation and Error Estimates
  • 2/22: Linear Finite Elements on [0,1]
  • 3/4: Error Estimates and Element Matrices (HW3 distributed)
  • 2/25: No Class, SIAM CSE meeting
  • 2/27: No Class, SIAM CSE meeting
  • 3/1: No Class, SIAM CSE meeting
  • 3/6: Function Spaces and Norms
  • 3/8: Lp and Sobolev Spaces
  • 3/11: G+ block Hilbert Spaces, Dual Spaces, and Riesz Representation
  • 3/13: G+ block Coercivity and Continuity, Lax-Milgram and Céa's Lemmas (HW4 distributed)
  • 3/15: Poincaré-Friedrichs Inequality, Element Choice
  • Spring Break!
  • 3/25: Approximation Properties, Non-elliptic problems
  • 3/27: Least-Squares Finite Elements
  • 3/29: Direct vs. Iterative Methods
  • 4/1: Jacobi and Gauss-Seidel (HW5 distributed)
  • 4/3: Incomplete Factorizations
  • 4/5: Polynomial Methods
  • 4/8: Arnoldi Algorithm and GMRES
  • 4/10: Lanczos and Conjugate Gradient
  • 4/12: Cost of CG; analyzing Jacobi
  • 4/17: Coarse-grid correction, two-grid algorithm (HW6 distributed)
  • 4/19: Multigrid and the V-cycle
  • 4/22: How effective is multigrid?
  • 4/24: Final Project Presentations
  • 4/26: Final Project Presentations
  • 4/29: Final Project Presentations
  • 5/8: HW6 and final project paper DUE