Quasi-interpolatory splines based on Schoenberg points

Abstract

Abstract. By using the Schoenberg points as quasi-interpolatory points, we achieve both generality and economy in contrast to previous sets, which achieve either generality or economy, but not both. The price we pay is a more complicated theory and weaker error bounds, although the order of convergence is unchanged. Applications to numerical integration are given and numerical examples show that the accuracy achieved, using the Schoenberg points, is comparable to that using other sets.

An adaptive IgA‐BEM with hierarchical B‐splines based on quasi‐interpolation quadrature schemes

“…In the remainder of the section we present the results of two numerical experiments aimed to corroborate Proposition 1 for the estimates e and E in (28). Convergence of the error with respect to h ℓi is studied on a sequence of uniform meshes.…”

“…The rules are effective also for nearly singular integrals. Although the use of spline QI for numerical integration was already studied in several papers, its introduction in the IgA context is a novelty.…”

“…The rule is effective also for nearly singular integrals. Although the use of spline quasi-interpolation for numerical integration was already studied in several papers [25,26,27,28], its introduction in the IgA context is a novelty. In particular, in this work, the QI formulas derived in [23] and [24] are integrated in a Galerkin BEM model and suitably framed into a hierarchical adaptive scheme.…”

Cubic Spline Approximation for Weakly Singular Integral Models

“…with a suitable choice of the nodes for the remaining values of : in i 3 T T 5  as in [7] and in as in [13]. 6 Let now consider the operator…”

“…arises from a compromise between two practical different constraints: maximizing the polynomial precision of the approximation and minimizing the collocation system order (see [13]). …”