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We introduce Gaussian quadrature rules for spline spaces that are frequentlyused in Galerkin discretizations to build mass and stiffness matrices. Bydefinition, these spaces are of even degrees. The optimal quadrature rules werecently derived [5] act on spaces of the smallest odd degrees and, therefore,are still slightly sub-optimal. In this work, we derive optimal rules directlyfor even-degree spaces and therefore further improve our recent result. We useoptimal quadrature rules for spaces over two elements as elementary buildingblocks and use recursively the homotopy continuation concept described in [6]to derive optimal rules for arbitrary admissible number of elements. Wedemonstrate the proposed methodology on relevant examples, where we deriveoptimal rules for various even-degree spline spaces. We also discussconvergence of our rules to their asymptotic counterparts, these are theanalogues of the midpoint rule of Hughes et al. [16], that are exact andoptimal for infinite domains.

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