A posteriori error estimation for the Laplace-Beltrami equation on spheres with spherical splines

We prove a posteriori upper and lower bounds for the error estimates when solving the Laplace-Beltrami equation on the unit sphere by using the Galerkin method with spherical splines. Adaptive mesh refinements based on these a posteriori error estimates are used to reduce complexity and computational cost of the corresponding discrete problems. The theoretical results are corroborated by numerical experiments.