Victor Manuel Calo

Michael Bartoň, Victor Manuel Calo

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.

Michael Bartoň, Rachid Ait-Haddou, Victor Manuel Calo

We provide explicit expressions for quadrature rules on the space of $C^1$ quintic splines with uniform knot sequences over finite domains. The quadrature nodes and weights are derived via an explicit recursion that avoids an intervention of any numerical solver and the rule is optimal, that is, it requires the minimal number of nodes. For each of $n$ subintervals, generically, only two nodes are required which reduces the evaluation cost by $2/3$ when compared to the classical Gaussian quadrature for polynomials. Numerical experiments show fast convergence, as $n$ grows, to the "two-third" quadrature rule of Hughes et al. for infinite domains.

Michael Bartoň, Victor Manuel Calo

We introduce a new concept for generating optimal quadrature rules for splines. Given a target spline space where we aim to generate an optimal quadrature rule, we build an associated source space with known optimal quadrature and transfer the rule from the source space to the target one, preserving the number of quadrature points and therefore optimality. The quadrature nodes and weights are, considered as a higher-dimensional point, a zero of a particular system of polynomial equations. As the space is continuously deformed by modifying the source knot vector, the quadrature rule gets updated using polynomial homotopy continuation. For example, starting with $C^1$ cubic splines with uniform knot sequences, we demonstrate the methodology by deriving the optimal rules for uniform $C^2$ cubic spline spaces where the rule was only conjectured heretofore. We validate our algorithm by showing that the resulting quadrature rule is independent of the path chosen between the target and the source knot vectors as well as the source rule chosen.

Rachid Ait-Haddou, Michael Bartoň, Victor Manuel Calo

We provide explicit expressions for quadrature rules on the space of $C^1$ cubic splines with non-uniform, symmetrically stretched knot sequences. The quadrature nodes and weights are derived via an explicit recursion that avoids an intervention of any numerical solver and the rule is optimal, that is, it requires minimal number of nodes. Numerical experiments validating the theoretical results and the error estimates of the quadrature rules are also presented.