Fig. 3.

(a) Representative solution $u=P_6(\cos \theta )$. Such functions are spherical harmonics which are analytic solutions to the Laplace equation in spherical coordinates. (b) High-order mesh of a patch of the unit sphere given by $\Omega =\{S^2(r,\theta ,\varphi )|(\theta ,\varphi )\in [-\frac{\pi }{4},\frac{\pi }{4}]\times [-\frac{\pi }{2},\frac{\pi }{2}],r=1\}$. This example consists of 36 triangular elements. The equivalent quadrilateral mesh consists of 18 elements obtained through recombining pairs of triangles.