Geometry dependence of the dimension of spaces of piecewise polynomials on rectilinear partitions Original Research Article

We consider the vector space of piecewise polynomials of total degree d and global smoothness order r on a certain type of rectilinear partition. The dimension of the space is shown to depend on the geometric relationship of a set of edges that are not all attached to one vertex. The geometric dependence persists even when d is large relative to r. This resultcontrasts with earlier results for spaces of piecewise polynomials on triangulations where, for d ⩾ 3r + 2, the dimension of the space is a function of the graph of the partition and the number of edges with different slopes attached to each interior vertex.