An Application of the Non-conforming Crouzeix-raviart Finite Element Method to Space Charge Calculations

The calculation of space charge effects in linear accelerators is an important prerequisite to understand the interaction between charged particles and the surrounding environment. These calculations should be as efficient as possible. In this work we explore the suitability of the CrouzeixRaviart Finite Element Method for the computation of the self-field of an electron bunch. INTRODUCTION Current and future accelerator design requires efficient 3D space charge calculations. One possible approach to Space Charge Calculations is the Particle-in-Cell (PIC) method, especially the Particle-Mesh method which calculates the potential in the rest-frame of the bunch. This computation usually is done by solving Poisson’s equation on the domain Ω, using a charge weighting f(x): −∆ u(x) = f(x), ∀x ∈ Ω. This equation is subject to some boundary conditions: u(x) = gD(x), ∀x ∈ ∂ΩD, grad u(x)·n(x) = gN (x), ∀x ∈ ∂ΩN . These computations should be as efficient as possible. SPACE CHARGE CALCULATIONS We are aiming at computing the self-field of the bunch. Denoting with D the dielectric flux and with ρ the charge density we are estimating a solution to Gauss’ law: