A Novel Galerkin Method for Solving PDEs on the Sphere Using Highly   Localized Kernel Bases
  We present a novel Galerkin method for solving partial differential equations on the sphere. The problem is discretized by a highly localized basis which is easily constructed. The stiffness matrix entries are computed by a recently developed quadrature formula unique to the localized bases we consider. We present error estimates and investigate the stability of the discrete stiffness matrix. Implementation and numerical experiments are discussed. 
F. J. Narcowich, Stephen T. Rowe, and Joseph D. Ward
math.NA