Fast discrete Helmholtz-Hodge decompositions in bounded domains
Abstract
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.
Domains
Mathematics [math] Analysis of PDEs [math.AP] Mathematics [math] Numerical Analysis [math.NA] Physics [physics] Physics [physics] Fluid Dynamics [physics.flu-dyn] Physics [physics] Physics [physics] Computational Physics [physics.comp-ph] Engineering Sciences [physics] Electromagnetism Engineering Sciences [physics] Mechanics [physics.med-ph] Fluids mechanics [physics.class-ph] Physics [physics] Mechanics [physics] Fluid mechanics [physics.class-ph]
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