On the dimension of bivariate superspline spaces

Abstract:

A bivariate piecewise polynomial function of total degree d on some grid partition $\Delta$ that has rth order continuous partial derivatives everywhere may have higher-order partial derivatives at the vertices of the grid partition. In finite element considerations and in the construction of vertex splines, it happens that only those functions with continuous partial derivatives of order higher than r at the vertices are needed to give the same full approximation order as the entire space of piecewise polynomials. This is certainly the case for $d \geq 4r + 1$. Such piecewise polynomial functions are called supersplines. This paper is devoted to the study of the dimension of certain superspline spaces. Since an exact dimension would have to depend on the geometric structure of the partition $\Delta$, we will give only upper and lower bounds. We will show, however, that the lower bound value is sharp for all quasi-crosscut partitions; and under suitable assumptions on r and d, the upper and lower bounds agree on both type-1 and type-2 arbitrary triangulations. In addition, a dimension criterion which guarantees that the lower bound gives the actual dimension is given.