A quadrupole ion trap (also known as Paul trap) is a device that con nes charged particles using dynamic electric elds produced by four voltage-varying electrodes. The aim of this project is to simulate numerically the movement of such ions and therefore design the optimal ion trap. In order to do so, a number of simulations were made from di erent approaches: by direct discretization of the di erential equations with the nite element method (FEM), and also by discretizing the integral equation using the method of moments (MoM), both in 2D and 3D. The voltage applied to the electrodes presents a harmonic variation, which takes place at a frequency that is low enough to consider a quasi-static problem, fact that simpli es the resolution. 2015/2016