Nada/CSC 2D1255 Spring 2007


Lab 1, multigrid for the Poisson equation: Report due Feb 23,

Computing the capacitance between an inner conductor an an enclosing square.
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The task is to solve the Laplace equation on the annular domain between an inner rectangle and an  outer  square.
Download the mglab text-file from the home page; it contains a skeleton
of the code you will develop.
The code does not  run at all as is. The solutions, driving functions, etc., are represented as arrays and we do not expand them into vectors.

1. Write a function u = jacobi2(u,rhs,mask)
which returns the result of one explicit time-step and the central difference
approximation to the laplace operator. Only the points where mask is non-zero 
should be updated. The array mask exists and every non-zero element is equal
to the relaxation parameter.
The equation is
    du/dt = laplace(u) + rhs
            u = 0 on outer square,
	    u = 1 on inner  rectangle
where rhs is a driving function, = 0 for the problem to be solved. But
you will need rhs non-zero in the multi-grid implementation so
should include it now.
Test the code by running the main program rcapmg with method = "Jacobi".

2. How does the convergence rate depend on the meshsize?
The theory says
           e(n+1)/e(n) = 1- kh**2

3. The convergence rate may be fast in the first iterations but
soon slows down. 
Try different initial values (hint: oscillatory or smooth are different)
Please explain

4. The computation of the capacitance by the line integral can use
any curve, as explained in the lecture from the discrete conservation
properties. The given code gauss uses
the outermost square. Make the code use
a smaller rectangular curve, and check that it does produce the same
answer. Or not? maybe the solution is not well enough converged.

5. Now try an elliptic inner conductor. Run a mesh refinement exercise and plot the convergence rate of the computed capacitance. What is the order of accuracy?
Hit: DO NOT USE too fine grids because the Jacobi iteration is so slow!

6. Now we turn to the multigrid exercise.
The skeleton code given is  patterned after the
V-cycle algorithm (see the Briggs copy.)
You need to 
1. construct the restriction and prolongation matrices.
2. figure out how to scale the residual when it is sent to the coarser
grid. The scaling depends on how you chose the scaling of the
matrices in the jacobi2-function!
When the implementation is correct you should observe
convergence rate independent of the meshsize, and no slowing down
after the initial steps.

7. The cheating here is that MG as implemented here does NOT
work properly for general geometries. Try the ellipse and see!
Can you explain why, and also give a characterization of the geometries
for which it does work properly , depending on the meshsize
and the number of MG levels? Construct the mask for a
non-trivial example (the 1/2 x 1/2 rectangle is trivial) and check that
your prediction is good.

8. (if there is time) Implement the FMV algorithm and check
the conclusions in Briggs that it DOES converge fatser.