An introduction to the numerical analysis of isogeometric methods

This paper gives an introduction to isogeometric methods from a mathematical point of view, with special focus on some theoretical results that are part of the mathematical foundation of the method. The aim of this work is to serve as a complement to other existing references in the field, that are more engineering oriented, and to provide a reference that can be used for didactic purposes. We analyse variational techniques for the numerical resolutions of PDEs using isogeometric methods, that is, based on splines or NURBS, and we provide optimal approximation and error estimates for scalar elliptic problems. The theoretical results are demonstrated by some numerical examples. We also present the definition of structure-preserving discretizations with splines, a generalization of edge and face finite elements, also with approximation estimates and some numerical tests for time harmonic Maxwell equations in a cavity.