A quasi-local interpolation operator�preserving the discrete divergence

Abstract

We construct an operator that preserves the discrete divergence and has the same quasi-local approximation properties as a regularizing interpolant; this is very useful when discretizing nonlinear incompressible fluid models. For low-degree finite elements, such operators have an explicit expression, from which local approximation properties can be easily derived. But for higher-degree finite elements, an explicit expression is generally not available and this construction is achieved by proving a global discrete inf–sup condition while using only local arguments. We write this construction in a general case, for conforming and non-conforming elements, and then give some applications.