Clement-type interpolation on spherical domains - interpolation error estimates and application to a posteriori error estimation

In this paper, a mixed boundary value problem for the Laplace-Beltrami operator is considered for spherical domains in R-3, i.e. for domains on the unit sphere. These domains are parametrized by spherical coordinates (phi, theta). A suitable finite element space is introduced. It corresponds to an isotropic triangulation of the underlying domain on the unit sphere. Error estimates are proven for a Clement-type interpolation operator, where appropriate weighted norms are used. The estimates are applied to the derivation of a reliable and efficient residual error estimator for the Laplace-Beltrami operator.