Criteria for the approximation property for multigrid methods in nonnested spaces

We extend the abstract frameworks for the multigrid analysis for nonconforming finite elements to the case where the assumptions of the second Strang lemma are violated. The consistency error is studied in detail for finite element discretizations on domains with curved boundaries. This is applied to prove the approximation property for conforming elements, stabilized -elements, and nonconforming elements for linear elasticity on nonpolygonal domains.