On C2 cubic quasi-interpolating splines and their computation by subdivision via blossoming

We discuss the construction of C2 cubic spline quasi-interpolation schemes defined on a refined partition. These schemes are reduced in terms of degrees of freedom compared to those existing in the literature. Namely, we provide a rule for reducing them by imposing super-smoothing conditions while preserving full smoothness and cubic precision. In addition, we provide subdivision rules by means of blossoming. The derived rules are designed to express the B-spline coefficients associated with a finer partition from those associated with the former one."Maria de Maeztu" Excellence Unit IMAG (University of Granada, Spain) CEX2020-001105-MICIN/AEI/10.13039/501100011033University of Granada University of Granada/CBU

Gaussian quadrature for C1C^1 cubic Clough-Tocher macro-triangles

A numerical integration rule for multivariate cubic polynomials over n-dimensional simplices was designed by Hammer and Stroud [14]. The quadrature rule requires n + 2 quadrature points: the barycentre of the simplex and n + 1 points that lie on the connecting lines between the barycentre and the vertices of the simplex. In the planar case, this particular rule belongs to a two-parameter family of quadrature rules that admit exact integration of bivariate polynomials of total degree three over triangles. We prove that this rule is exact for a larger space, namely the C1 cubic Clough-Tocher spline space over macro-triangles if and only if the split-point is the barycentre. This results into a factor of three reduction in the number of quadrature points needed to integrate the Clough-Tocher spline space exactly

Shock capturing computations with stabilized Powell-Sabin elements

International audienceIn the last recent years, thanks to the increasing power of the computational machines , the interest in more and more accurate numerical schemes is growing. Methods based on high-order approximations are nowadays the common trend in the computational research community, in particular for CFD applications. This work is focused on Powell-Sabin (PS) finite elements, a finite element method (FEM) based on PS splines. PS splines are piecewise quadratic polynomials with a global C1 continuity , defined on conforming triangulations. Despite its attractive characteristics, so far this scheme hasn't had the attention it deserves. PS splines are adapted to unstructured meshes and, contrary to classical tensor product B-splines, they are particularly suited for local refinement , a crucial aspect in the analysis of highly convective and anisotropic equations. The additional global smoothness of the C1 space has a beneficial stabilization effect in the treatment of advection-dominated equations and leads to a better capturing of thin layers. Finally, unlike most of other typology of high-order finite elements, the numerical unknowns in PS elements are located in the vertices of the triangulation, leading to an easy treatment of the parallel aspects. Some geometrical issues related to PS elements are discussed here, in particular, the generation of the control triangles and the imposition of the boundary conditions. The PS FEM method is used to solve the compressible Euler equation in supersonic regime. A classical shock-capturing technique is used to reduce the oscillation around the discontinuity, while a variational multiscale formulation is used to introduce numerical diffusion in the streamwise direction. Some typical numerical examples are used to evaluate the performance of the PS discretization