MATH 272: NUMERICAL METHODS FOR PDE

Instructor: Randolph E. Bank

  • The class meets MWF, 1:00-1:50, AP&M 2402.
  • Office hours are 4:00-5:00 Monday and Wednesday, AP&M 5771.
  • While the lectures will not be taken directly from any textbook, there are a number of books that can be used as references.
    • Numerical Approximations of Partial Differential Equations, by Alfio Quarteroni and Alberto Valli. (Springer Series in Computational Mathematics, Volume 23, ISBN 978-3-540-85268-1) Electronic Version.
    • PDEs with Numerical Methods, by Stig Larsson and Vidar Thomee. (Springer Texts in Applied Mathematics, Volume 45, ISBN 978-3-540-01772-1, hardcover, and 978-3-540-88705-8, softcover) Electronic Version.
    • The Mathematical Theory of Finite Element Methods by Susanne C. Brenner and L. Ridgway Scott (Springer Texts in Applied Mathematics, Volume 15, ISBN 978-0-387-75934-0) Electronic Version.

Math 272A -- Fall Quarter 2016
Basic Discretization Methods

  • Two point Boundary Value Problem
    • Review some basic theory (existence, uniqueness, stability...)
    • Finite difference methods, basic error estimates
    • Finite element methods, basic error estimates
  • Elliptic Equations in Two and Three Space Dimensions
    • Finite difference methods, basic error estimates
    • Finite element methods, basic error estimates
    • Finite volume (box) methods, basic error estimates
  • Introduction to Computational Fluid Dynamics
    • Convection-Diffusion Equations, upwinding, Petrov-Galerkin methods
    • Stokes and Navier-Stokes Equations
Homework 1 (Due Oct. 5)
Homework 2 (Due Oct. 12)
Homework 3 (Due Oct. 26)
Homework 3 Solutions
Homework 4 (Due Nov. 2)
Homework 5 (Due Nov. 16)
Homework 6 (Due Nov. 23)
Homework 7 (Due Nov. 30)

Math 272B -- Winter Quarter 2017
Basic Solution Methods

  • Iterative Methods for Sparse Systems of Equations
    • Classical Iterative Methods (Jacobi, Gauss-Seidel, SOR)
    • Conjugate Gradient, Biconjugate Gradient Methods
    • ILU and SSOR Preconditioners
    • Multigrid, Domain Decomposition and Hierarchical Basis Methods
  • Sparse Direct Methods
  • Nonlinear Equations
    • Newton's Method in the PDE Environment
    • Eigenvalue Problems
Homework 8 (Due Feb. 1)
Homework 2 (Due Feb. 8)
Homework 3 (Due Feb. 22)
Homework 4 (Due Mar. 8)
Homework 5 (Due Mar. 15)

Math 272C -- Spring Quarter 2017
Time Dependent Problems and Adaptive Methods

  • Basic Time Discretization Schemes
    • Introduction to Stiff ODE's
    • Method of lines discretization, basic error estimates
    • Space-time finite element methods, moving mesh methods
  • Introduction to Adaptive Methods
    • A Posteriori error estimates
    • Adaptive mesh refinement algorithms
    • Adaptive mesh moving algorithms
    • h-p adaptive finite element methods