Circle/Disk
The first harmonic function on a unit disk \(\Omega = {(x,y): x^2+y^2\leq1}\) is \[ u_\text{exact}(r\cos\varphi, r\sin\varphi) = J_0(\beta_{0,1} r), \hspace{1em} r\in[0,1], \] where \(J_0\) is the Bessel function of the first kind and \(\beta_{0,1}\approx2.4048\) is its first zero. There is radial symmetry in this eigenfunction so the angular component \(\varphi\) in polar coordinates does not appear explicitly.