Numerical solution of partial differential equations with Powell-Sabin splines

Powell-Sabin splines are piecewise quadratic polynomials with global C-continuity. They are defined on conformal triangulations of two-dimensional domains, and admit a compact representation in a normalized B-spline basis. Recently, these splines have been used successfully in the area of computer-aided geometric design for the modelling and fitting of surfaces. In this paper, we discuss the applicability of Powell-Sabin splines for the numerical solution of partial differential equations defined on irregular domains. A Galerkin-type PDE discretisation is derived, and elaborated for the variable coefficient diffusion equation. Special emphasis goes to the treatment of Dirichlet and Neumann boundary conditions. Finally, an error estimator is developed and an adaptive mesh refinement strategy is proposed. We will illustrate the effectiveness of the approach by means of some numerical experiments.