HAL CCSD Multiscaled asymptotic expansions for the electric potential: Surface charge densities and electric fields at rounded corners Ciarlet, Patrick Kaddouri, Samir Propagation des Ondes : Étude Mathématique et Simulation (POEMS) ; Inria Saclay - Ile de France ; Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA) ; École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS) International audience ISSN: 0218-2025 EISSN: 1793-6314 Mathematical Models and Methods in Applied Sciences World Scientific Publishing hal-00876233 https://hal-ensta-paris.archives-ouvertes.fr/hal-00876233 https://hal-ensta-paris.archives-ouvertes.fr/hal-00876233 Mathematical Models and Methods in Applied Sciences, 2007, 17 (6), pp.845-876. &#x27E8;10.1142/s0218202507002133&#x27E9; DOI: 10.1142/s0218202507002133 info:eu-repo/semantics/altIdentifier/doi/10.1142/s0218202507002133 en [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] info:eu-repo/semantics/article Journal articles We are interested in computing the charge density and the electric field at the rounded tip of an electrode of small curvature. As a model, we focus on solving the electrostatic problem for the electric potential. For this problem, Peek's empirical formulas describe the relation between the electric field at the surface of the electrode and its curvature radius. However, it can be used only for electrodes with either a purely cylindrical, or a purely spherical, geometrical shape. Our aim is to justify rigorously these formulas, and to extend it to more general, two-dimensional, or three-dimensional axisymmetric, geometries. With the help of multiscaled asymptotic expansions, we establish an explicit formula for the electric potential in geometries that coincide with a cone at infinity. We also prove a formula for the surface charge density, which is very simple to compute with the Finite Element Method. In particular, the meshsize can be chosen independently of the curvature radius. We illustrate our mathematical results by numerical experiments. © World Scientific Publishing Company. 2007