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An isogeometric approach for solving the Laplace-Beltrami equation on atwo-dimensional manifold embedded in three-dimensional space using a Galerkinmethod based on Catmull-Clark subdivision surfaces is presented and assessed.The scalar-valued Laplace-Beltrami equation requires only C0 continuity and isadopted to elucidate key features and properties of the isogeometric methodusing Catmull-Clark subdivision surfaces. Catmull-Clark subdivision bases areused to discretise both the geometry and the physical field. A fitting methodgenerates control meshes to approximate any given geometry with Catmull-Clarksubdivision surfaces. The performance of the Catmull-Clark subdivision methodis compared to the conventional finite element method. Subdivision surfaceswithout extraordinary vertices show the optimal convergence rate. However,extraordinary vertices introduce error, which decreases the convergence rate. Acomparative study shows the effect of the number and valences of theextraordinary vertices on accuracy and convergence. An adaptive quadraturescheme is shown to reduce the error.

Assessment of an Isogeometric Approach with Catmull-Clark Subdivision Surfaces using the Laplace-Beltrami Problems