During the quiz you may use your textbook, my notes, and 
your own notes.

No calculators or other electronic devices are permitted.

Please make sure your cell phones are quiet during class and
off during quizzes.

12-22-2010: Sorry for leaving in this next paragraph.
Given the sample questions below, it obviously
doesn't apply.

This quiz will only cover material in 
"Solution of ODEs, BVPs, Part 1" of the course notes,
and the corresponding material in the textbook.
Part 2 of the notes will not be included.

For Quiz 3, be able to:


-- Construct a finite difference or finite element approximation
   to a given ODE-BVP.

-- Use the discrete maximum principle for ODE-BVPs.

-- Show that if a(u,phi_j) = (f, phi_j) for j = 1,...,M-1,
   then a(u,v) = (f,v) for all v in S_h.

-- Use the error formulas for FD/FE approximations
   to ODE-BVPs to draw conclusions about how
   fast the error will decrease.

-- Use Theorem 5.2.

-- Suppose we change the boundary conditions from u(0)=u(1)=0
   to u'(0)=u'(1)=0.  Assume that b(x) = 0 and c(x) > 2
   for x in [0,1].  The solution exists.

      Write the weak formulation.
      (Use test functions v in H^1.)

      What can you conclude about existence and uniqueness of
      the solution to the weak problem?

      Write the linear systems corresponding to
      finite difference and finite element approximations
      to this problem.  What can you say about existence
      and uniqueness of the discrete solutions?  How close
      are they to the solution to the ODE?

-- Compare and contrast finite difference and finite element
   methods for solving ODE-BVPs.  For example:

    -- Which requires less work to form and solve the
       linear system?

    -- Which should be used if we want to compute an
       approximation to u'?

    -- For each method, what can we do if we decide
       that we need to reduce the error? (reduce h?
       increase the order of approximation?)

-- Apply the Maximum Principle and the Minimum Principle
   to draw conclusions about the behavior of the solution
   to an elliptic PDE.

-- Use Theorem 3.2.

-- Define uniqueness and stability of the solution of an elliptic
   PDE