Abstract
In this work, we study quasi-interpolation in a space of sextic splines defined over Powell–Sabin triangulations. These spline functions are of class C2 on the whole domain but fourth-order regularity is required at vertices and C3 regularity is imposed across the edges of the refined triangulation and also at the interior point chosen to define the refinement. An algorithm is proposed to define the Powell–Sabin triangles with a small area and diameter needed to construct a normalized basis. Quasi-interpolation operators which reproduce sextic polynomials are constructed after deriving Marsden’s identity from a more explicit version of the control polynomials introduced some years ago in the literature. Finally, some tests show the good performance of these operators.
Highlights
Spline functions over triangulations have been, and are, the object of intense research for their role in the Approximation Theory and for dealing with a wide variety of problems of practical interest, among which the approximation of scattered data and the numerical solution of partial differential equations occupy a prominent place.It is well known that the requirement of a regularity Cm of a spline on a given triangulation implies that the degree must be greater than or equal to 4m + 1 [1]
It is essential to use splines of the lowest degree for a given class, different finite elements obtained by subdiving every triangle have been introduced and analyzed in the literature, among them the Clough–Tocher (CT-) and Powell–Sabin (PS-) refinements
A first subdivision into six triangles is achieved by selecting an inner point in every triangle and connecting it with similar points in the adjacent triangles as well as with the three vertices
Summary
Spline functions over triangulations have been, and are, the object of intense research for their role in the Approximation Theory and for dealing with a wide variety of problems of practical interest, among which the approximation of scattered data and the numerical solution of partial differential equations occupy a prominent place. By using the relationship between polynomials and their blossoms, we obtained a result that will allow to define the control polynomial that was the main tool for establishing Marsden’s identity which is the key for building quasi-interpolation schemes based on a C2 sextic PS-spline space. Let V := {Vi}in=v1, Z := {Zi}in=t 1 and E ∗ be, respectively, the subsets of vertices in ∆, split points in ∆PS, and edges in ∆PS that connect a split point Zi to a point Rij. As given in [26], the space of PS splines is defined as. Non-negative and locally supported basis functions Biv,j and Bkt with respect to vertices and triangles, respectively, that form a partition of unity were defined, and any s ∈ S62,4,3(Ω, ∆PS) could be represented as nv 15 nt s(x, y) := ∑ ∑ civ,j Biv,j(x, y) + ∑ ctk Bkt (x, y). They are used to define the B-splines Biv,j and Bkt as follows
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
