Divergence-free quasi-interpolation

Abstract

Divergence-free interpolation has been extensively studied and widely used in approximating vector-valued functions that are divergence-free. However, so far the literature contains no treatment of divergence-free quasi-interpolation. The aims of this paper are two-fold: to construct an analytically divergence-free quasi-interpolation scheme and to derive its simultaneous approximation orders to both the approximated function and its derivatives. To this end, we first explicitly construct a divergence-free matrix kernel based on polyharmonic splines and study its properties both in the spatial domain and Fourier domain. Then, with this divergence-free matrix kernel, we construct a divergence-free quasi-interpolation scheme defined in the whole space R d for some positive integer d . We also derive corresponding approximation orders of quasi-interpolation to both the approximated divergence-free function and its derivatives. Finally, by coupling divergence-free interpolation together with our divergence-free quasi-interpolation, we extend our construction to a divergence-free quasi-interpolation scheme defined over a bounded domain. Numerical simulations are presented at the end of the paper to demonstrate the validity of divergence-free quasi-interpolation.