Anisotropic H1-Stable Projections on Quadrilateral Meshes

Projections of H functions onto finite element spaces are of fundamental interest in numerical analysis. In particular, they are necessary to prove stability and a priori estimates of stabilized finite element schemes. Examples of such projections on isotropic elements are the Clement interpolation [4], and the variant of Scott and Zhang [6] in order to maintain Dirichlet conditions. For an overview of anisotropic interpolation operators we refer to the book [1] where several H-stable projections on tensor grids are addressed. In this work we use a projection operator Bh : H(Ω) → Qh developed in [2] which is suitable for anisotropic quadrilateral meshes obtained by bilinear transformations. Originally it was designed and analyzed for meshes aligned with the coordinate axes. In this work, this interpolation operator is considered for a much general class of quadrilateral meshes, which introduces further couplings between the partial derivatives. In particular, we allow for bilinear transformations, where the nonlinear contribution and the shearing should be limited by the mesh sizes into the particular directions of anisotropy. The results of this work are used in [3]. We consider anisotropic meshes without any restriction of grid alignment with the coordinate axis. The transformation TK from the reference cells K to the physical cell K is allowed to be bilinear. Such a transformation can be expressed as a composition of translation, rotation, shearing and stretching, augmented with the pure bilinear term,