  An isogeometric approach for solving the Laplace-Beltrami equation on a
two-dimensional manifold embedded in three-dimensional space using a Galerkin
method based on Catmull-Clark subdivision surfaces is presented and assessed.
The scalar-valued Laplace-Beltrami equation requires only C0 continuity and is
adopted to elucidate key features and properties of the isogeometric method
using Catmull-Clark subdivision surfaces. Catmull-Clark subdivision bases are
used to discretise both the geometry and the physical field. A fitting method
generates control meshes to approximate any given geometry with Catmull-Clark
subdivision surfaces. The performance of the Catmull-Clark subdivision method
is compared to the conventional finite element method. Subdivision surfaces
without extraordinary vertices show the optimal convergence rate. However,
extraordinary vertices introduce error, which decreases the convergence rate. A
comparative study shows the effect of the number and valences of the
extraordinary vertices on accuracy and convergence. An adaptive quadrature
scheme is shown to reduce the error.
