p23.m

% p23.m - eigenvalues of perturbed Laplacian on [-1,1]x[-1,1]
% (compare p16.m)
% Set up tensor product Laplacian and compute 4 eigenmodes:
N = 16; [D,x] = cheb(N); y = x;
[xx,yy] = meshgrid(x(2:N),y(2:N)); xx = xx(:); yy = yy(:);
D2 = D^2; D2 = D2(2:N,2:N); I = eye(N-1);
L = -kron(I,D2) - kron(D2,I);
% Laplacian
L = L + diag(exp(20*(yy-xx-1))); % to remvoe perturbation ~ f, comment this line
% + perturbation
[V,D] = eig(L); D = diag(D);
[D,ii] = sort(D); ii = ii(1:4); V = V(:,ii);
% Reshape them to 2D grid, interpolate to finer grid, and plot:
[xx,yy] = meshgrid(x,y);
fine = -1:.02:1; [xxx,yyy] = meshgrid(fine,fine);
uu = zeros(N+1,N+1);
[ay,ax] = meshgrid([.56 .04],[.1 .5]); clf
for i = 1:4
uu(2:N,2:N) = reshape(V(:,i),N-1,N-1);
uu = uu/norm(uu(:),inf);
uuu = interp2(xx,yy,uu,xxx,yyy,'cubic');
subplot('position',[ax(i) ay(i) .38 .38])
contour(fine,fine,uuu,-.9:.2:.9)
colormap([0 0 0]), axis square
title(['eig = ' num2str(D(i)/(pi^2/4),'%18.12f') '\pi^2/4'])
end