Weighted quadrature for hierarchical B-splines

Abstract

We present weighted quadrature for hierarchical B-splines to address the fast formation of system matrices arising from adaptive isogeometric Galerkin methods with suitably graded hierarchical meshes. By exploiting a local tensor-product structure, we extend the construction of weighted rules from the tensor-product to the hierarchical spline setting. The proposed algorithm has a computational cost proportional to the number of degrees of freedom and advantageous properties with increasing spline degree. To illustrate the performance of the method and confirm the theoretical estimates, a selection of 2D and 3D numerical tests is provided.