Numerical Analysis of Boundary Value Problems

Course Description

Textbook

• Course Notes for 585-6 by R.J. LeVeque.
  Will be made available.
  We will use Chapters 1-5 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.
• Handouts will be distributed on some additional topics.

Syllabus (and tentative schedule)

Numerical interpolation and differentiation (2 lectures)

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

Function space methods

• Spectral methods
• Finite element method

Multidimensional boundary value problems for PDEs (3 weeks)

Laplace/Poisson equation - some physical examples

Finite difference method; solution via Gaussian elimination.

Fast Poisson solvers using FFT.

Iterative solution of sparse matrix problems (3 weeks)

• Jacobi, Gauss-Seidel, SOR
• Conjugate gradient and preconditioning
• Methods for nonsymmetric systems
• Multigrid

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 every week or two 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

Follow links in the table below to obtain a copy of the homework in PostScript (.ps) or Adobe Acrobat (.pdf) format. You may also obtain here solutions to some of the homework and exam problems. An item shown below in plain text is not yet available. For additional information regarding viewing and printing the homework and solution sets, click here.

Tentative Schedule

	Date	Event	Homework Problem Sets
Week 1	W, Jan. 4	First day of classes
Week 2	F, Jan. 13	Homework 1 due	hw1
Week 3	M, Jan. 16	Martin Luther King Day
Week 4	F, Jan. 27	Homework 2 due	hw2
Week 7	M, Feb. 13	Homework 3 due	hw3
	F, Feb. 17	Midterm	Review sheet
Week 8	M, Feb. 20	President's Day
Week 9	F, March 3	Homework 4 due	hw4
Finals week	W, March 15	Final project due	project

Grading

Homework: 50%, midterm exam: 25%, final project: 25%.
There will probably be 4 homework assignments, with due dates as indicated in the schedule above.

You may view your homework and exam grades on-line.

Other resources

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