Given a vector field $X^a$ on a sphere, I want to decompose it into a surface divergence-free component and a surface curl-free component (similar to the Helmholtz decomposition, but on the 2-dimensional sphere $S^2$ instead of $\mathbb{R}^3$). The problem is described quite well here:
http://www.chebfun.org/examples/sphere/HelmholtzDecomposition.html
So apparently, for a tangential vector field such a decomposition exists, and the problem of finding this decomposition is equivalent to solving Poisson's equation on the sphere. However, I would like to have an analytical solution, but all I have found so far are descriptions for how to solve this problem numerically. Are there any analytical solutions? I.e., given the function $f$ on the sphere $S^2$, can I obtain an explicit form for the function $\Phi$ fulfilling the equation $\nabla^2\Phi=f$?
