Finite element quasi-interpolation and best approximation

This paper introduces a quasi-interpolation operator for scalar- and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces. This operator gives optimal estimates of the best approximation error in any L p -norm assuming regularity in the fractional Sobolev spaces W r,p , where p ∈ [ 1,∞ ] and the smoothness index r can be arbitrarily close to zero. The operator is stable in L 1 , leaves the corresponding finite element space point-wise invariant, and can be modified to handle homogeneous boundary conditions. The theory is illustrated on H 1 -, H (curl)- and H (div)-conforming spaces.