Gauss-Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis

TL;DR

This paper develops optimal Gaussian quadrature rules for even-degree spline spaces used in isogeometric analysis, improving previous sub-optimal rules and enabling efficient numerical integration in finite and infinite domains.

Contribution

The authors derive new optimal quadrature rules directly for even-degree spline spaces, enhancing previous methods and providing a recursive construction approach for multiple elements.

Findings

Derived optimal rules for various even-degree spline spaces

Demonstrated recursive construction using homotopy continuation

Showed convergence to asymptotic quadrature rules for infinite domains

Abstract

We introduce Gaussian quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. By definition, these spaces are of even degrees. The optimal quadrature rules we recently derived [5] act on spaces of the smallest odd degrees and, therefore, are still slightly sub-optimal. In this work, we derive optimal rules directly for even-degree spaces and therefore further improve our recent result. We use optimal quadrature rules for spaces over two elements as elementary building blocks and use recursively the homotopy continuation concept described in [6] to derive optimal rules for arbitrary admissible number of elements. We demonstrate the proposed methodology on relevant examples, where we derive optimal rules for various even-degree spline spaces. We also discuss convergence of our rules to their asymptotic counterparts, these are the analogues of the midpoint rule of Hughes et al. [16], that are exact and optimal for infinite domains.