Fast discrete Helmholtz–Hodge decompositions in bounded domains

We present new fast discrete Helmholtz–Hodge decomposition (DHHD)   methods for efficiently computing at the order O(ε)O(ε) the divergence-free (solenoidal) or curl-free (irrotational) components and their associated potentials for a given L2(Ω) vector field in a bounded domain. The solution algorithms solve suitable penalized boundary-value elliptic problems involving either the grad(div)grad(div) operator in the vector penalty-projection (VPP)   or the rot(rot) operator in the rotational penalty-projection (RPP) with adapted right-hand sides   of the same form. Therefore, they are extremely well-conditioned, fast and cheap, avoiding having 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 εε is sufficiently small. We state optimal error estimates vanishing as O(ε)O(ε) with a penalty parameter εε as small as desired up to machine precision, e.g. ε=10−14ε=10−14. Some numerical results confirm the efficiency of the proposed (DHHD) methods, very useful for solving problems in electromagnetism or fluid dynamics.