Bases for C0-P1 divergence-free elements and for C1-P2 finite elements on union jack grids

It is a challenge to find point wise (including inter-element boundary) divergence-free finite element bases. By identifying functions in the kernel of the divergence operator, we discover a local basis for the full divergence-free space of the C0-P1 finite element, on the union jack grid. The optimal order of approximation is shown for the P1 divergence-free finite elements on union jack grids, and numerical tests are provided, solving stationary Stokes equations. We further compute the anti-derivative of such divergence-free basis functions to construct a C1-P2 basis for the continuously-differentiable, piecewise quadratic polynomial space on union jack grids. The full approximation property of the C1-P2 space is established. The C1-P2 basis is applied to the biharmonic equation. The optimal order of convergence is proved. Numerical tests are presented to support the analysis.