Interpolation in h‐Version Finite Element Spaces

Interpolation operators associate with a function an element from a finite element space. The difference between the two functions is called interpolation error. Estimates of the interpolation error are used for a priori and a posteriori estimation of the discretization error of finite element methods. For a priori error estimates, one can often use the nodal interpolant. To get optimal error bounds, the maximum available regularity of the solution has to be used. The situation is different for a posteriori error estimates. For the investigation of residual-type error estimators, local interpolation error estimates for functions from the Sobolev space W1, 2(Ω) are needed. Such estimates can be obtained for many finite elements only for quasi-interpolation operators, where point values of functions or derivatives are replaced in the definition of the interpolation operators by certain averaged values. This paper gives an overview of different interpolation operators and their error estimates. The discussion restricts the h-version of the finite element method, but it includes interpolation on the basis of triangular/tetrahedral and quadrilateral/hexahedral meshes, affine and nonaffine elements, isotropic and anisotropic elements, and Lagrangian and other elements. Keywords: nodal interpolation; quasi-interpolation; Clement interpolation; Scott–Zhang interpolation; affine mapping; isoparametric mapping; isotropic finite element; shape-regular element; anisotropic element