Quadrature rules in the isogeometric Galerkin method: State of the art and an introduction to weighted quadrature

In this paper we introduce the quadrature needed in the isogeometric Galerkin method. Quadrature rules affect the cost of the assembly of the discrete counterpart of the IGA method, so that the search for efficient quadrature is an active research topic. The focus of the first part is on a brief survey on the contributions available for the reduction of computational costs for such issue. We review the generalized Gaussian strategies and the reduced quadrature. Then we present the novelty of weighted quadrature, recently proposed by Calabrò, Sangalli and Tani for the efficient assembly. We detail the construction of such rules and give some examples. Finally, we end with some remarks on current work and further developments.