  Subdivision surfaces are proven to be a powerful tool in geometric modeling
and computer graphics, due to the great flexibility they offer in capturing
irregular topologies. This paper discusses the robust and efficient
implementation of an isogeometric discretization approach to partial
differential equations on surfaces using subdivision methodology. Elliptic
equations with the Laplace-Beltrami and the surface bi-Laplacian operator as
well as the associated eigenvalue problems are considered. Thereby, efficiency
relies on the proper choice of a numerical quadrature scheme which preserves
the expected higher order consistency. A particular emphasis is on the
robustness of the approach in the vicinity of extraordinary vertices. In this
paper, the focus is on Loop's subdivision scheme on triangular meshes. Based on
a series of numerical experiments, different quadrature schemes are compared
and a mid-edge quadrature, which is easy-to-implement via lookup tables, turns
out to be a preferable choice due to its robustness and efficiency.
