function U = lp_lexact(kind, X, Y)
% LP_LEXACT  Exact solutions on the L-shaped domain (-1,1)^2 minus the quadrant
% x > 0, y < 0.
%
% The re-entrant corner sits at the origin, with interior angle 3*pi/2 between
% the edge y = 0, x > 0 and the edge x = 0, y < 0. Measuring the angle from the
% first of those, counterclockwise, the domain is 0 < theta < 3*pi/2, and
%
%     s(r, theta) = r^(2/3) sin(2 theta / 3)
%
% is harmonic there and vanishes on both edges of the corner. It is the slowest
% decaying member of the family r^(2k/3) sin(2k theta/3) that a general
% Dirichlet datum excites, and it is the reason the corner is a problem: s is
% continuous, but its gradient is infinite at the origin and it has no second
% derivative there, so every convergence rate derived from a Taylor expansion
% is void.
%
%   'sing'   the singular function alone
%   'smooth' log|z - z0| with z0 outside the domain: harmonic, and smooth right
%            up to the corner, so the corner costs nothing
%   'both'   the sum, which is the generic case
z0 = 1.6 + 1.3i;
th = atan2(Y, X);
th(th < -1e-13) = th(th < -1e-13) + 2*pi;      % theta in [0, 3*pi/2] on the domain
r = sqrt(X.^2 + Y.^2);
sing = r.^(2/3) .* sin(2*th/3);
smooth = 0.5 * log((X - real(z0)).^2 + (Y - imag(z0)).^2);

switch kind
    case 'sing'
        U = sing;
    case 'smooth'
        U = smooth;
    otherwise
        U = sing + smooth;
end
end