A two-level local projection stabilisation on uniformly refined triangular meshes

The two-level local projection stabilisation on triangular meshes is based on the refinement of a macro cell into three child cells by connecting the barycentre with the vertices of the macro cell. This refinement technique leads to non-nested meshes with large inner angles and to non-nested finite element spaces. We show that also the red refinement where a triangle is divided into four child cells by connecting the midpoints of the edges can be used. This avoids the above mentioned disadvantages. For the red refinement a local inf-sup condition for the continuous, piecewise polynomial approximation spaces of order less than or equal to r ≥ 2 living on the refined mesh and discontinuous, piecewise polynomial projection spaces of order less than or equal to r − 1 living on the coarser mesh is established. Numerical tests compare the local projection stabilisation resulting from both refinement rules in case of convection-diffusion problems.