Carlos J. García-Cervera.
South Hall, 6707.
Office Phone: (805) 893 3681
Office Fax: (805) 893 2385
Office Hours: Tuesday and Thursday, 12:30-2:00pm

• Numerical Solution of Partial Differential Equations by the Finite Element Method, by Claes Johnson.
• Computational Differential Equations, by by K. Eriksson, D. Estep, P. Hansbo, and C. Johnson.
• Finite elements for electrical engineers, by Peter P. Silvester and Ronald L. Ferrari.
• The Mathematical Theory of Finite Element Methods, by Susanne C. Brenner and L. Ridgway Scott.
• A Multigrid Tutorial, by William L. Briggs, Van Emden Henson, and Steve F. McCormick.
• Multilevel Adaptive Methods for Partial Differential Equations, by Steve F. McCormick.
• Numerical Solution of Partial Differential Equations, by K.W. Morton and D.F. Mayers.
• Finite Element Method: Its basis and Fundamentals, by O.C. Zienkiewicz, R.L. Taylor, and J.Z. Zhu.

For details of the implementation, the following book can be used:

• Understanding and Implementing the Finite Element Method, by Mark S. Gockenbach (available from SIAM).

You can also find some FEM resources at the following:

• Internet Finite Element Resources.
• FEM for Magnetics.
• Finite Element Center.

For a review of Advanced Calculus, you may want to check out the following:

• Vector calculus, by Jerrold E. Marsden and Anthony J. Tromba.

Prerequisites: Knowledge of a computer language suitable for numerical computing: FORTRAN, C, C++, or Matlab. Previous knowledge in Numerical Analysis is recommended, although not necessary.

Assignments and grading: Homework will be assigned every two weeks. It will be handed out on Fridays, and will be collected at the beginning of the class on the Friday of the second following week. Late homeworks will not be accepted. The homework will generally consist of some theoretical questions, and some computational assignments. You will be required to write a program to solve certain problems. The program must be given to me as part of the assignment, together with the output of the program, in the format indicated in the assignment, and an interpretation of the results whenever necessary. You can write the programs either in FORTRAN, C, C++, or Matlab.

In addition to the homework, there will be a final project. Your final grade for the course will be decided according to the following formula:

Final Grade = 60% Homework + 40% Final

You should try and do a search on the Internet, since there are a lot of sites dedicated to Matlab, and programming in general.

Professor Douglas Arnold, at the Institute for Mathematics and its Applications, has a list of disaters due to bad numerical computing. Check it out here!

I wrote the syllabus and the homework assignments using a program called LaTeX. If you want to learn more about this program, you can find some information and tutorials at the following homepages:

• The LaTeX Project.
• Text processing using LaTeX.

Syllabus: (Available also in postscript and pdf). During this course we will try to follow the following schedule. However, much like everything said earlier, this is subject to change.

Week 1: Introduction to the FEM for elliptic problems.

1. Variational formulation of a one-dimensional model problem.
2. FEM for the model problem with piecewise linear functions.
3. An error estimate for FEM for the model problem.
4. FEM for the Poisson equation.
5. The Hilbert spaces , , and .
6. A geometric interpretation of FEM.
7. A Neumann problem. Natural and essential boundary conditions.
8. Remarks on programming.
9. Homework 1: pdf. Due on Thursday, October 19th, 2006.

Week 2: Abstract formulation of the finite element method for elliptic problems.

1. Introduction. The continuous problem.
2. Discretization. An error estimate.
3. The energy norm.

Weeks 3 & 4: Some finite element spaces and approximation theory for FEM.

1. Introduction. Regularity requirements.
2. Some examples of finite elements.
3. Interpolation with piecewise linear functions in two dimensions.
4. Interpolation with polynomials of higher degree.
5. Error estimates for FEM for elliptic problems.
6. On the regularity of the exact solution.

Week 5: Some applications to elliptic problems.

1. The elasticity problem.
2. Stokes problem.
3. Maxwell's equations.

Weeks 6 & 7: Solving linear systems of equations.

1. Gaussian elimination. Cholesky's method.
2. Condition number.
3. Iterative methods: Jacobi and Gauss-Seidel.
4. The conjugate Gradient method.
5. Preconditioning.
6. Multigrid methods.

Week 8: FEM for parabolic problems.

1. A one dimensional model problem.
2. Semi-discretization in space.
3. Discretization in space and time.
4. Numerical solution of ODEs.
5. The discontinuous Galerkin Method.

Week 9: Boundary element methods.

1. Some integral equations.
   1. An integral equation for an exterior Dirichlet problem using a single layer potential.
   2. An exterior Dirichlet problem with double layer potential.
   3. An exterior Neumann problem with single layer potential.
   4. Alternative integral equation formulations.
2. Finite element methods.
   1. FEM for a Fredholm equation of the first kind.
   2. FEM for a Fredholm equation of the second kind.

Week 11: FEM for hyperbolic problems.

1. Transport in one dimension.
2. The wave equation in one dimension.
3. The wave equation in higher dimensions.
4. A finite element method.
5. Error estimates and adaptive error control.