Abstract
Several approximation operators followed Philippe Clément’s seminal paper in 1975 and are hence known as Clément-type interpolation operators, weak-, or quasi-interpolation operators. Those operators map some Sobolev space V ⊂ W k,p(Ω) onto some finite element space V h ⊂ W k,p(Ω) and generalize nodal interpolation operators whenever W k,p(Ω) ⊄ C 0(Ω), i.e., when p ≤ n/k for a bounded Lipschitz domain Ω ⊂ ℝn. The original motivation was H 2 ⊄ C 0(Ω) for higher dimensions n ≥ 4 and hence nodal interpolation is not well defined.Todays main use of the approximation operators is for a reliability proof in a posteriori error control. The survey reports on the class of Clément type interpolation operators, its use in a posteriori finite element error control and for coarsening in adaptive mesh design.KeywordsPosteriori ErrorError ControlPosteriori Error EstimationInterpolation OperatorDual NormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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