Prerequisites: Math 228A or Math 128A, or equivalent background in numerical analysis.

Text(s):

1. Keith Miller, Math 228B notes on Sobolev spaces, linear elliptic equations and finite element methods. Xerox copies will be available first day in class.

2. Claes Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press.

3. Randall J. LeVeque, Numerical Methods for Conservation Laws, Birkhauser Verlag.

Grading: 60% on problem sets due every 2 weeks. 40% on a 3-hour final exam.

  From (1) we will study Sobolev spaces, weak solutions of linear uniformly elliptic equations, the Galerkin or finite element method for their numerical solution, and also the Galerkin method for nonlinear elliptic and parabolic problems.

  From (2) we will study a wider variety of finite element topics on elliptic and parabolic problems, although at a more elementary level. We will also consider an introduction to the streamline diffusion finite element method for hyperbolic problems in Chapter 9.

  From (3) we will study finite difference methods for nonlinear hyperbolic conservation laws. Chapters 1-3 consider weak solutions, shocks, rarefaction waves, and the entropy condition. Chapters 10-11 review methods for linear problems, especially with nonsmooth data. Chapters 12-17 consider methods for nonlinear conservation laws (especially scalar conservation laws in 1-D) including Godunov methods, monotone, TVD and ENO schemes.

  We will also include, from my own papers, an introduction to the Moving Finite Element method in which nodes are allowed to concentrate and move automatically with the sharp moving fronts.