Clement interpolation and its role in adaptive finite element error control

Several approximation operators followed Philippe Clement's seminal paper in 1975 and are hence known as Clement-type interpolation operators, weak-, or quasi-interpolation operators. Those operators map some Sobolev space V subset of W-k,W-p(Omega) onto some finite element space V-h subset of W-k,W-p(Omega) and generalize nodal interpolation operators whenever W-k,W-p(Omega) not subset of C-0((Omega) over bar), i.e., when p <= n/k for a bounded Lipschitz domain Omega subset of R-n. The original motivation was H-2 not subset of C-0(Omega) 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 Clement type interpolation operators, its use in a posteriori finite element error control and for coarsening in adaptive mesh design.