Numerical Analysis of Boundary Value Problems

Course Description

Textbook

• Finite Difference Methods for Ordinary and Partial Differential Equations by R.J. LeVeque, SIAM 2007.
  We will use Chapters 1-4 and the Appendices this quarter.
• Some references on iterative methods:
  • A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM, 1997.
  • L.N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, 1997.
  • Loyce's Conjugate Gradient Notes
  • Loyce's Preconditioned Conjugate Gradient Notes
  • Loyce's FFT Discrete Sine Transform Notes
• Handouts will be distributed on some additional topics.

Syllabus (and tentative schedule)

Chapter 1. Finite difference approximations (2 lectures)

Chapter 2. Two-point boundary value problems for ODEs (3 weeks)

Character of solution; boundary conditions.

Finite difference method for linear problem u'' = f, BCs at x = 0,1.

Accuracy, convergence, stability.

Finite difference methods for nonlinear problems

Boundary layers and nonuniform grids

Chapter 3. Elliptic Equations (3 weeks)

Laplace/Poisson equation - some physical examples

Finite difference method; solution via Gaussian elimination.

Fast Poisson solvers using FFT.

Chapter 4. Iterative Methods for Sparse Linear Systems (2 weeks)

Jacobi, Gauss-Seidel, SOR

Conjugate gradient and preconditioning

Methods for nonsymmetric systems

Multigrid methods

Learning Objectives and Instructor Expectations

The course will be a combination of computation and theoretical analysis. The goal is to obtain an understanding of numerical methods and their implementation, as well as learning mathematical techniques for analyzing the stability and accuracy of these methods.

There will be homework assignments roughly bi-weekly that will involve MATLAB programming and written exercises. You may consult with your classmates about how to do the homework, but you should write your own code and express the answers to the written questions in your own words.

Schedule and Homework

Grading

Other resources

Tutorials