L2-orthogonal projections onto finite elements on locally refined meshes are H1-stable

Abstract

We merge and extend recent results which prove the H1-stability of the L2-orthogonal projection onto standard finite element spaces, provided that the underlying simplicial triangulation is appropriately graded. For lowest-order Courant finite elements S1(T) in Rd with d>=2, we prove that such a grading is always ensured for adaptive meshes generated by newest vertex bisection. For higher-order finite elements Sp(T) with p>=1, we extend existing bounds on the polynomial degree with a computer-assisted proof. We also consider L2-orthogonal projections onto certain subspaces of Sp(T) which incorporate zero Dirichlet boundary conditions resp. an integral mean zero property.