Isogeometric analysis is a numerical simulation method based on the NURBS
(non-uniform rational B-spline) representation of CAD (computer aided design)
models. The NURBS are directly used to generate the finite-dimensional space of
test functions which is then used for the physical simulation. More precisely, the
isogeometric test functions are obtained by composing the inverse of the geometry
mapping with the NURBS basis functions. The special case of singularly
parametrized NURBS surfaces and NURBS volumes is used to represent
non-quadrangular or non-hexahedral domains without splitting, which leads to a
very compact and convenient representation. We analyze the influence of
singularities on several properties of the isogeometric discretization.

In the presence of singularities in the parametrization, some of the resulting
test functions are not well defined. Thus they do not necessarily possess the
required regularity properties. Consequently, numerical methods for the solution
of partial differential equations can not be applied properly. Moreover,
approximation properties of isogeometric function spaces depend on the regularity
of the parametrization. Furthermore, evaluation of the classical basis functions
near the singularity may not be stable.
 
In this talk we will study the regularity and smoothness properties of the
isogeometric test functions for various classes of singularly parametrized
domains. In addition we discuss the influence of inner control points on the
regularity of the test function space. We prove bounds for the approximation error
for two classes of singular parametrizations of planar domains. Moreover, we
present local refinement strategies that lead to geometrically regular splittings
of singular patches.