École Nationale Supérieure de
Techniques Avancées

Abstract

First, we focus on solving numerically the Poisson problem with homogenous Dirichlet conditions in a three dimensional prismatic or axisymmetric domain, with a reentrant edge at the boundary. We present the Fourier Singular Complement Method based on a Fourier expansion in the direction parallel to the reentrant edge and the Singular Complement Method for solving the 2D problems in the Fourier modes. The analysis shows that we recover the optimal rate of convergence O(h) when using P 1 Lagrange finite elements for the discretization. No refinement near the reentrant edge is required in the computations. Second, we are interested in computing the charge density and the electricfield at the rounded tip of an electrode of small curvature radius. Our model problem is the electrostatic problem. 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, they apply only to thin electrodes with either a purely cylindrical, or a purely spherical, geometrical shape. Our aim is to justify rigorously these formulas, and to extend them to more general, either two dimensional or three dimensional axisymmetric, geometries. With the help of multiscaled asymptotic expansions, we establish rigorously 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 FE Method. In particular, the meshsize can be chosen independently of the curvature radius. We illustrate our mathematical results by numerical experiments.