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T. Apel, J. Melenk:
"Interpolation and quasi-interpolation in h- and hp-version finite element spaces (extended version)";
in: "ASC Report 39/2015", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2015, ISBN: 978-3-902627-09-4, 1 - 60.

Interpolation operators map a function u to an element Iu of a finite element space. Unlike more general approximation operators, interpolants are defined locally. Estimates of the interpolation error,
., the difference u − Iu, are of utmost importance in numerical analysis. These estimates depend on the size of the finite elements, the polynomial degree employed, and the regularity of u. In contrast to interpolation the term quasi-interpolation is used when the regularity is so low that interpolation has to be combined with regularization. This paper gives an overview of different interpolation operators and their error estimates. The discussion includes the h-version and the hp-version of the finite element method, interpolation on the basis of triangular/tetrahedral and quadrilateral/hexahedral meshes, affine and nonaffine elements, isotropic and anisotropic elements, and Lagrangian and other elements.
approximation, polynomial interpolation, nodal interpolation, quasi-interpolation, Cl´ement interpolation, Scott-Zhang interpolation, isotropic finite element, shape-regular element, anisotropic element, h-version, p-version, hp-version

http://www.asc.tuwien.ac.at/preprint/2015/asc39x2015.pdf