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 L¹, leaves the corresponding finite element space point-wise invariant, and can be modified to handle homogeneous boundary conditions. The theory is illustrated on H¹-, H(curl)- and H(div)-conforming spaces.