Proof of linear independence of flat-top PU-based high-order approximation

This paper extends a rank deficiency counting approach, which was initially established by An et al. (2011, 2012 [1,2]) to determine the rank deficiency of finite element partition of unity (PU)-based approximations, to explicitly prove the linear independence of the flat-top PU-based high-order polynomial approximation. The study also examines the coupled flat-top PU and finite element PU-based approximation, and the results indicate that the space at a global level is also linearly independent for 1-D setting and 2-D setting with triangular mesh, but not so for rectangular mesh. Moreover, a new procedure is proposed to simplify the construction of flat-top PU, and its feasibility, accuracy and efficiency have been validated by a typical numerical example.