Hierarchical generating systems that are derived from Zwart–Powell (ZP) elements can be used to generate quadratic splines on adaptively refined criss-cross triangulations. We propose two extensions of these hierarchical generating systems, firstly decoupling the hierarchical ZP elements, and secondly enriching the system by including auxiliary functions. These extensions allow us to generate the entire hierarchical spline space – which consists of all piecewise quadratic C 1 -smooth functions on an adaptively refined criss-cross triangulation – if the triangulation fulfills certain technical assumptions. Special attention is dedicated to the characterization of the linear dependencies that are present in the resulting enriched decoupled hierarchical generating system. 
Completeness of generating systems for quadratic splines on adaptively refined criss-cross triangulations