mit18086_smoothing.m

function mit18086_smoothing
%MIT18086_SMOOTHING
%    Sets up a 1d Poisson problem with an oscillatory right hand
%    side and applies various iterative solvers to it.
%    Observe the decay rate of various error frequencies:
%    Jacobi smooths medium frequencies best, but fails for
%    grid-scale oscillations. Weighted-Jacobi remedies this
%    problem at the expense of a slower decay rate for small
%    frequencies. Gauss-Seidel performs well as a smoother.
%    SOR smooths best, but it is also more expensive in practice.
%    Observe how the error gets smoothed very quickly, but the
%    actual convergence to the correct solution is very slow.

% 04/2008 by Benjamin Seibold
% Feel free to modify for teaching and learning.
%------------------------------------------------------------------
b = 11;                           % number of oscillations
n = 50;                           % number of grid points
%------------------------------------------------------------------
% set up 1d Poisson problem with oscillatory right hand side
h = 1/(n+1);
x = (h:h:(1-h))';
sx = 2*pi*b*x.^2;
f = 1e2*sx.*sin(sx);
A = spdiags(ones(n,1)*[-1 2 -1],-1:1,n,n);
b = f*h^2;
uc = A\b;
I = speye(n);
D = diag(diag(A));
L = tril(A)-D;
%------------------------------------------------------------------
% different preconditioners
name = {'Jacobi','weighted Jacobi','Gauss-Seidel','SOR'};
P = {D,3/2*D,D+L,D+1.8*L};
%------------------------------------------------------------------
np = length(P);
for k = 1:np
    M{k} = I-P{k}\A;
    c{k} = P{k}\b;
    u{k} = zeros(n,1);
end
for i = 1:100
    clf
    for k = 1:np, u{k} = M{k}*u{k}+c{k}; end
    subplot(1,2,1)
    plot(x,uc,'k.-')
    hold all, for k = 1:np, plot(x,u{k},'.-'), end, hold off
    title(sprintf('solution after %d steps',i))
    subplot(1,2,2)
    hold all, for k = 1:np, plot(x,uc-u{k},'.-'), end, hold off
    legend(name)
    title(sprintf('error after %d steps',i))
    drawnow
end