Anisotropic interpolation error estimates via orthogonal expansions (Q1945340)

This paper defines an anisotropic interpolation operator on a reference square considering polynomial spaces of arbitrary degree. Its construction is based on orthogonal expansions with respect to Lobatto polynomials. An interpolation property is then proved. A standard technique [\textit{T. Apel}, Anisotropic finite elements: Local estimates and applications. Advances in Numerical Mathematics, Leipzig: Teubner (1999; Zbl 0934.65121), Theorem 2.6] then implies an interpolation result for a family of affine equivalent (parallelogram) meshes satisfying (i) a maximum angle condition and (ii) a coordinate system condition: the sine of the angle between the longest side and the \(x\)-axis is bounded by a constant and the aspect ratio. Note that such a family of meshes does not need necessarily to be regular. This result is then generalized to three dimensions and a similar anisotropic interpolation result for Lagrange hexahedral (parallelepiped) elements is proved.