  Non Uniform Rational B-spline (NURBS) patches are a standard way to describe
complex geometries in Computer Aided Design tools, and have gained a lot of
popularity in recent years also for the approximation of partial differential
equations, via the Isogeometric Analysis (IGA) paradigm. However, spectral
accuracy in IGA is limited to relatively small NURBS patch degrees (roughly p
<= 8), since local condition numbers grow very rapidly for higher degrees. On
the other hand, traditional Spectral Element Methods (SEM) guarantee spectral
accuracy but often require complex and expensive meshing techniques, like
transfinite mapping, that result anyway in inexact geometries. In this work we
propose a hybrid NURBS-SEM approximation method that achieves spectral accuracy
and maintains exact geometry representation by combining the advantages of IGA
and SEM.
  As a prototypical problem on non trivial geometries, we consider the
Laplace--Beltrami and Allen--Cahn equations on a surface. On these problems, we
present a comparison of several instances of NURBS-SEM with the standard
Galerkin and Collocation Isogeometric Analysis (IGA).
