[NA] Andrea Bressan: Dimension of piecewise polynomials on the Wang-Shi macroelement

16 February 2024 12:30 till 13:15 - Location: Timmanzaal LB.01.170 | Add to my calendar

In dimension one the construction of piecewise polynomials with given degree, number of continuous derivatives and the  subdomains is a solved problem. Already in dimension two the rank of the continuity condition depends on the geometry of the polygonal subdomains and thus the space dimension can only be computed case by case. This changes if the subdomains are sufficiently structured for the required pair of degree and number of continuous derivatives. On cartesian meshes C^k splines are achieved with any degree d>k using a tensor product construction and both the space dimension and its properties are easily derived by the univariate case. On any triangulation C^0 splines can be constructed for all degree>0. More generally C^k functions on a general triangulation can be achieved for degree d>= 3k+2, e.g. for k=1, d=5 is the Argyris element. (a local basis requires d>=4k+1) An alternative is to add additional structure to the partition by replacing a the triangles of a triangulation with macro-elements, i.e. by “splitting” the triangles of a triangulation in subdomains. Examples are C^1-cubic on the Clough–Tocher split and C^2-quintics on the Powell-Sabin 12 split. Recently C^2-cubics (2022) and C^3-quartics (2024) spaces have been constructed using the Wang-Shi split where the macroelements consist of polygons with possibly many vertices. The talk will summarize an “elementary" proof that the dimension of C^(d-1) splines of degree d on the Wang-Shi macroelement can be expressed in purely combinatorial terms.