Subdivision-based isogeometric analysis for second order partial differential equations on surfaces

We investigate the isogeometric analysis approach based on the extended Catmull-Clark subdivision for solving the PDEs on surfaces. As a compatible technique of NURBS, subdivision surfaces are capable of the refinability of B-spline techniques, and overcome the major difficulties of the interior parameterization encountered by the isogeometric analysis. The surface is accurately represented as the limit form of the extended Catmull-Clark subdivision, and remains unchanged throughout the h-refinement process. The solving of the PDEs on surfaces is processed on the space spanned by the Catmull-Clark subdivision basis functions. In this work, we establish the interpolation error estimates for the limit form of the extended Catmull-Clark subdivision function space on surfaces. We apply the results to develop the approximation estimates for solving multiple second-order PDEs on surfaces, which are the Laplace-Beltrami equation, the Laplace-Beltrami eigenvalue equation and the time-dependent Cahn-Allen equation. Numerical experiments confirm the theoretical results and are compared with the classical linear finite element method to demonstrate the performance of the proposed method.