A structure-preserving finite element framework for the Vlasov-Maxwell system

TL;DR

This paper introduces a new finite element framework that preserves the structure of the Vlasov-Maxwell system, employing high-order elements and a novel stabilization technique, achieving optimal convergence and validating on challenging benchmarks.

Contribution

A novel, structure-preserving finite element method with a residual-based artificial viscosity for the Vlasov-Maxwell system, ensuring stability, high-order accuracy, and effective benchmark performance.

Findings

Achieved optimal convergence orders for all polynomial spaces tested.

Successfully solved challenging benchmark problems.

Validated the robustness and accuracy of the proposed method.

Abstract

We present a stabilized, structure-preserving finite element framework for solving the Vlasov-Maxwell equations. The method uses a tensor product of continuous polynomial spaces for the spatial and velocity domains, respectively, to discretize the Vlasov equation, combined with curl- and divergence-conforming N\'ed\'elec and Raviart-Thomas elements for Maxwell's equations on Cartesian grids. A novel, robust, consistent, and high-order accurate residual-based artificial viscosity method is introduced for stabilizing the Vlasov equations. The proposed method is tested on the 1D2V and 2D2V reduced Vlasov-Maxwell system, achieving optimal convergence orders for all polynomial spaces considered in this study. Several challenging benchmarks are solved to validate the effectiveness of the proposed method.