Fast discrete Helmholtz-Hodge decompositions in bounded domains

We present new fast {\em discrete Helmholtz-Hodge decomposition (DHHD)} methods to efficiently compute at the order $\cO(\eps)$ the divergen\-ce-free (solenoidal) or curl-free (irrotational) components and their associated potentials of a given $\mathbf{L}^2(\Omega)$ vector field in a bounded domain. The solution algorithms solve suitable penalized boundary-value elliptic problems involving either the $\Grad(\Div)$ operator in the {\em vector penalty-projection (VPP)} or the $\Rot(\Rot)$ operator in the {\em rotational penalty-projection (RPP)} with {\em adapted right-hand sides} of the same form. Therefore, they are extremely well-conditioned, fast and cheap avoiding to solve the usual Poisson problems for the scalar or vector potentials. Indeed, each (VPP) or (RPP) problem only requires two conjugate-gradient iterations whatever the mesh size, when the penalty parameter $\varepsilon$ is sufficiently small. We state optimal error estimates vanishing as $\mathcal{O}(\varepsilon)$ with a penalty parameter $\varepsilon$ as small as desired up to machine precision, e.g. $\varepsilon=10^{-14}$. Some numerical results confirm the efficiency of the proposed (DHHD) methods, very useful to solve problems in electromagnetism or fluid dynamics.