We merge and extend recent results which prove the H1-stability of theL2-orthogonal projection onto standard finite element spaces, provided that theunderlying simplicial triangulation is appropriately graded. For lowest-orderCourant finite elements S1(T) in Rd with d>=2, we prove that such a grading isalways ensured for adaptive meshes generated by newest vertex bisection. Forhigher-order finite elements Sp(T) with p>=1, we extend existing bounds on thepolynomial degree with a computer-assisted proof. We also considerL2-orthogonal projections onto certain subspaces of Sp(T) which incorporatezero Dirichlet boundary conditions resp. an integral mean zero property.
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