  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.
