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

The two-level local projection stabilization with the pair (Qr,h,Qr−1,2hdisc), r≥1, of spaces of continuous, piecewise (mapped) polynomials of degree r on the mesh Th in each variable and discontinuous, piecewise (mapped) polynomials of degree r−1 on the macro mesh Mh 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 (Qr,h,Qr,2hdisc), r≥2, with the enriched projection space Qr,2hdisc 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 (Qr,h,Qr−1,2hdisc) in higher order cases r≥2.