Sparse tensor product spectral Galerkin BEM for elliptic problems with random input data on a spheroid

We introduce and analyze a sparse tensor product spectral Galerkin Boundary Element Method based on spherical harmonics for elliptic problems with random input data on a spheroid. Problems of this type appear in geophysical applications, in particular in data acquisition by satellites. Aiming at a deterministic computation of the k-th order statistical moments of the random solution, we establish convergence theorems showing that the sparse tensor product spectral Galerkin discretization is superior to the full tensor product spectral Galerkin discretization in the case of mixed regularity of the data’s k-th order moments, naturally implying mixed regularity of the k-th order moments of the random solution. We prove that analytic regularity of the data’s k-th order moments implies analytic regularity of the solution’s k-th order moments. We illustrate performance of the sparse and full tensor product discretization schemes on several numerical examples.