NURBS in isogeometric discretization methods: A spectral analysis

Nonuniform rational B-splines (NURBS) are the most common representation form in isogeometric analysis. In this article, we study the spectral behavior of discretization matrices arising from isogeometric Galerkin and collocation methods based on d-variate NURBS of degrees (p1,...,pd), and applied to general second-order partial differential equations defined on a d-dimensional domain. The spectrum of these matrices can be compactly and accurately described by means of a so-called symbol. We compute this symbol and show that it is the same as in the case of isogeometric discretization matrices based on d-variate polynomial B-splines of degrees (p1,...,pd). The theoretical results are confirmed with a selection of numerical examples.