Tom Lyche

Andrea Bressan, Tom Lyche

This paper analyzes the approximation properties of spaces of piece-wise tensor product polynomials over box meshes with a focus on application to IsoGeometric Analysis (IGA). The errors are measured in Lebesgue norms. Estimates of different types are considered: local and global, with full or reduced Sobolev seminorms. Attention is also paid to the dependence on the degree and exponential convergence is proved for the approximation of analytic functions.

Tom Lyche, Georg Muntingh

We review the construction and a few properties of the S-basis, a simplex spline basis for the $C^1$ quadratic splines on the Powell-Sabin 12-split.

Tom Lyche, Georg Muntingh

For the space $\mathcal{S}$ of $C^3$ quintics on the Powell-Sabin 12-split of a triangle, we determine the simplex splines in $\mathcal{S}$ and the six symmetric simplex spline bases that reduce to a B-spline basis on each edge, have a positive partition of unity, a (barycentric) Marsden identity, and domain points with an intuitive control net. We provide a quasi-interpolant with approximation order 6 and a Lagrange interpolant at the domain points. The latter can be used to show that each basis is stable in the $L_\infty$ norm, which yields an $h^2$ bound for the distance between the Bézier ordinates and the values of the spline at the corresponding domain points. Finally, for one of these bases we provide $C^0$, $C^1$, and $C^2$ conditions on the control points of two splines on adjacent macrotriangles, and a conversion to the Hermite nodal basis.

Tom Lyche, Georg Muntingh

For the space of $C^3$ quintics on the Powell-Sabin 12-split of a triangle, we determine explicitly the six symmetric simplex spline bases that reduce to a B-spline basis on each edge, have a positive partition of unity, a Marsden identity that splits into real linear factors, and an intuitive domain mesh. The bases are stable in the $L_\infty$ norm with a condition number independent of the geometry, have a well-conditioned Lagrange interpolant at the domain points, and a quasi-interpolant with local approximation order 6. We show an $h^2$ bound for the distance between the control points and the values of a spline at the corresponding domain points. For one of these bases we derive $C^0$, $C^1$, $C^2$ and $C^3$ conditions on the control points of two splines on adjacent macrotriangles.