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Fast discrete Helmholtz-Hodge decompositions in bounded domains

Philippe Angot 1, * Jean-Paul Caltagirone 2, 3 Pierre Fabrie 4
* Corresponding author
1 analyse appliquée
LATP - Laboratoire d'Analyse, Topologie, Probabilités
4 Équipe EDP et Physique Mathématique
IMB - Institut de Mathématiques de Bordeaux
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.
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https://hal.archives-ouvertes.fr/hal-00756959
Contributor : Philippe Angot <>
Submitted on : Saturday, November 24, 2012 - 4:41:45 PM
Last modification on : Monday, August 31, 2020 - 9:36:04 AM
Long-term archiving on: : Monday, February 25, 2013 - 3:45:23 AM

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Philippe Angot, Jean-Paul Caltagirone, Pierre Fabrie. Fast discrete Helmholtz-Hodge decompositions in bounded domains. Applied Mathematics Letters, Elsevier, 2013, 26 (4), pp.445--451. ⟨10.1016/j.aml.2012.11.006⟩. ⟨hal-00756959⟩

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