Galerkin approximation for elliptic PDEs on spheres

We discuss a Galerkin approximation scheme for the elliptic partial differential equation −∆u+ ω2u = f on Sn ⊂ Rn+1. Here ∆ is the Laplace-Beltrami operator on Sn, ω is a non-zero constant and f belongs to C2k−2(Sn), where k ≥ n/4 + 1, k is an integer. The shifts of a spherical basis function φ with φ ∈ H τ (Sn) and τ > 2k ≥ n/2 + 2 are used to construct an approximate solution. An H1(Sn)error estimate is derived under the assumption that the exact solution u belongs to C2k(Sn).