function lu=lapmf(u)
% LAPMF
%
% matrix-free negative Laplacian
%
% function lu=lapmf(u)
%
% This function applies the 2D Laplacian to a vector in a matrix-free
% manner. To make it consistent with the solves, the input u is
% a linear array with dimension n*n, where n+1 MUST be a power of 2
% if you plan to use P2D in any way. You must use reshape to get to/from
% 2D n x n arrays defined on the geometric grid to the n*n x 1 arrays 
% that solvers expect. This code does the 1D -> 2D -> 1D conversion
% for you. 
%
n2=length(u);
n=sqrt(n2);
h2=(n+1)*(n+1);
%
% transform the linear array into a 2d array
%
uu=zeros(n,n);
uu(:)=u;
%
% compute the discrete negative Laplacian on the 2d array
%
luu=4*uu;
if n>1
luu(1:n-1,:)=luu(1:n-1,:)-uu(2:n,:);
luu(2:n,:)=luu(2:n,:)-uu(1:n-1,:);
luu(:,2:n)=luu(:,2:n)-uu(:,1:n-1);
luu(:,1:n-1)=luu(:,1:n-1)-uu(:,2:n);
end
%
% convert to a linear array, divide by h^2
%
lu=luu(:)*h2;