% Example 8: Laplace Neumann BVP on a starfish domain via single-layer
% representation, with the W (onesmat) regularization added so that the
% second-kind integral equation is invertible.

mip load --install magland/magland/chunkie;

tic;

% chunker discretization of a starfish domain
chnkr = chunkerfunc(@(t) starfish(t));

% boundary condition. by symmetry of the starfish this integrates to zero
pwfun = @(r) r(1,:).^2.*r(2,:);
rhs = pwfun(chnkr.r); rhs = rhs(:);

% kernel: normal derivative of the Laplace single-layer
kernsp = kernel('lap','sprime');

% matrix discretization of the BIE
sysmat = chunkermat(chnkr,kernsp);
sysmat = sysmat + 0.5*eye(chnkr.npt) + onesmat(chnkr);

% solve
sigma = gmres(sysmat,rhs);

% grid for plotting
x1 = linspace(-2,2,300);
[xx,yy] = meshgrid(x1,x1);
targs = [xx(:).'; yy(:).'];
in = chunkerinterior(chnkr,targs);
uu = nan(size(xx));

% evaluate solution using the single layer (not its derivative)
kerns = kernel('lap','s');
uu(in) = chunkerkerneval(chnkr,kerns,sigma,targs(:,in));

toc;

figure(1); clf
h = pcolor(xx,yy,uu); set(h,'EdgeColor','none'); colorbar
hold on; plot(chnkr,'k'); colormap(redblue)