Interpolation inh‐Version Finite Element Spaces

Abstract

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 W 1, 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.

Evaluating Local Approximations of the <em>L</em>²-Orthogonal Projection Between Non-Nested Finite Element Spaces

“…Whereas the original results in [26] have been derived for triangular meshes in case d = 2, the relevant properties can indeed be proved for finite element spaces associated with general, not necessarily affine meshes and d ∈ {2, 3}. Note that the assertion holds true for non-quasi-uniform meshes; we refer the reader to the discussion in [4]. The technical ideas of the proof are perhaps most clearly elaborated in [12,Lemma 3.1], although in a slightly different context.…”

“…We also learned about advanced techniques for the construction of transfer operators from [41,62,63] in the context of non-conforming domain decomposition methods. Other interesting studies giving basic insights into the analysis of approximation operators in finite element spaces, which influenced our work, can be found in [4,12,13,20,24,[56][57][58]64].…”