Interpolation inh‐Version Finite Element Spaces

Abstract

AbstractInterpolation operators associate with a function an element from a finite element space. The difference between the two functions is calledinterpolation 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 thenodal 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 spaceW1, 2(Ω) are needed. Such estimates can be obtained for many finite elements only forquasi‐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 theh‐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.