Quasi-interpolation based on the ZP-element for the numerical solution of integral equations on surfaces in $mathbb R^3$

The aim of this paper is to present spline methods for the numerical solution of integral equations on surfaces of R^3 , by using optimal superconvergent quasi-interpolants defined on type-2 triangulations and based on the Zwart–Powell quadratic box spline. In particular we propose a modified version of the classical collocation method and two spline collocation methods with high order of convergence. We also deal with the problem of approximating the surface. Finally, we study the approximation error of the above methods together with their iterated versions and we provide some numerical tests.