A two-level higher order local projection stabilization on hexahedral meshes

The two-level local projection stabilization with the pair ( Q r , h , Q r − 1 , 2 h disc ) , r ≥ 1 , of spaces of continuous, piecewise (mapped) polynomials of degree r on the mesh T h in each variable and discontinuous, piecewise (mapped) polynomials of degree r − 1 on the macro mesh M h in each variable satisfy a local inf–sup condition leading to optimal error estimates. In this note, we show that even the pair of spaces ( Q r , h , Q r , 2 h disc ) , r ≥ 2 , with the enriched projection space Q r , 2 h disc satisfies the local inf–sup condition and can be used in this framework. This gives a new, alternative proof of the inf–sup condition for the pair ( Q r , h , Q r − 1 , 2 h disc ) in higher order cases r ≥ 2 .