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 ) the divergence-free (solenoidal) or curl-free (irrotational) components and their associated potentials for a given L 2 ( Ω ) vector field in a bounded domain. The solution algorithms solve suitable penalized boundary-value elliptic problems involving either the 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 ε is sufficiently small. We state optimal error estimates vanishing as O ( ε ) with a penalty parameter ε as small as desired up to machine precision, e.g.  ε = 1 0 − 14 . Some numerical results confirm the efficiency of the proposed (DHHD) methods, very useful for solving problems in electromagnetism or fluid dynamics.