Triangle
The first harmonic function on a equilateral triangle with vertices \((0,0)\), \((1,0)\), \((1/2, \sqrt{3}/2)\) is \[ u_\text{exact}(x, y) = \sin\left(2\pi\cdot \frac{2y}{\sqrt{3}}\right) + \sin2\pi\left(x - \frac{y}{\sqrt{3}}\right) + \sin2\pi\left(1-x-\frac{y}{\sqrt{3}} \right), \hspace{1em} (x,y)\in\bigtriangleup. \] The corresponding eigenvalue is \(\lambda=\frac{16}{3}\pi^2\).