function U = lp_exact2d(kind, X, Y, par)
% LP_EXACT2D  A harmonic function on the unit square, used as ground truth.
%
% Every choice here satisfies Laplace's equation exactly, so taking its trace
% on the boundary as Dirichlet data makes it the exact solution of the
% Dirichlet problem, and the difference between a computed solution and this
% function is the whole of the error.  par is a per-kind shape parameter.
if nargin < 4
    par = 1;
end
switch kind
    case 'expsin'
        % exp(x) sin(y): entire, gentle, the standard test
        U = exp(X) .* sin(Y);
    case 'poly'
        % Re (x + iy)^k: a harmonic polynomial of degree k
        k = round(par);
        U = real((X + 1i*Y).^k);
    case 'wave'
        % exp(k*y) sin(k*x): oscillatory along one edge, growing across the
        % square by a factor exp(k), which is where its conditioning goes
        k = par;
        U = exp(k*Y) .* sin(k*X);
    case 'log'
        % log|z - z0| with z0 a distance par outside the right edge: harmonic
        % inside, but with a singularity just off the domain, so the solution
        % has structure on the length scale par near that edge
        z0 = (1 + par) + 0.5i;
        U = log(abs((X + 1i*Y) - z0));
    otherwise
        error('lp_exact2d: unknown solution "%s"', kind);
end
end