A primal formulation for the Helmholtz decomposition

In 1999, Jean-Paul Caltagirone and Jérôme Breil have developed in their paper [Caltagirone, J. Breil, Sur une méthode de projection vectorielle pour la résolution des équations de Navier–Stokes, C.R. Acad. Sci. Paris 327(Série II b) (1999) 1179–1184] a new method to compute a divergence-free velocity. They have used the grad(div) operator to extract the solenoidal part of a given vector field. In this contribution we explain how this method can be considered as a real Helmholtz decomposition and we present a stable approximation in the framework of spectral methods. Numerical results are presented to illustrate the efficiency of this approach.