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 R3, i.e. for domains on the unit sphere. These domains are parametrized by spherical coordinates (φ, 𝛉). 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 Clément-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.