In several applications, as e.g. for problems in Solid Mechanics, Fluid Dynamics,
and Biology, Partial Differential Equations (PDEs) are defined on surfaces. The
numerical approximation of the PDEs often involves an approximation of the
computational domain, represented by the surface, which may possibly lead to
significant errors. Since a wide range of surfaces of practical interest can be
represented by B-splines or NURBS, we consider the numerical solution of the PDEs by
means of Isogeometric Analysis, a numerical method based on the isoparametric
concept for which the same basis functions used to represent the computational
domain are then used for approximating the unknown solution field of the PDEs.
In this work, we solve linear, nonlinear, and time dependent PDEs involving the
second order Laplace-Beltrami or high-order operators on surfaces by means of
NURBS-based Isogeometric Analysis in the Galerkin framework. We highlight and
discuss the properties of the method in terms of accuracy, efficiency, and
regularity of the NURBS basis functions by performing error and spectrum analyses.