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.