Weighted quadrature for hierarchical B-splines

TL;DR

This paper introduces a weighted quadrature method for hierarchical B-splines that efficiently constructs system matrices in adaptive isogeometric Galerkin methods, maintaining computational efficiency and accuracy in higher dimensions.

Contribution

It extends weighted quadrature rules from tensor-product to hierarchical B-splines, enabling fast matrix assembly in adaptive isogeometric analysis.

Findings

Computational cost scales with degrees of freedom.

Method performs well for high spline degrees.

Numerical tests confirm theoretical efficiency.

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.