Clément Interpolation and Its Role in Adaptive Finite Element Error Control
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 Error Error Control Posteriori Error Estimation Interpolation Operator Dual Norm
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