A P-Adaptive Hybridizable Discontinuous Galerkin Spectral Element Method for Electrostatic Particle-in-Cell Simulations

Abstract

This paper presents a p-adaptive high-order hybridizable discontinuous Galerkin spectral element method (HDG-SEM) for solving the Poisson equation in electrostatic plasma simulations using particle-in-cell (PIC) schemes. This approach enables element-local refinement of the polynomial degree, concentrating computational effort specifically in regions with strong gradients. Thus, the method significantly reduces the global number of degrees of freedom compared to uniform high-order methods. The proposed method is implemented in the open-source framework PICLas and validated through a series of benchmark test cases, including a dielectric sphere and a one-dimensional plasma sheath. Finally, a two-dimensional axisymmetric simulation of an ion optic demonstrates the method's capability to efficiently model complex plasma phenomena but also highlights current limitations.

Figures (7)

• Figure 1: Cutaway showing the simulated absolute value of the electric field and the edges of the curved mesh.
• Figure 2: Refinement levels and P-adaptation area used for the dielectric sphere convergence tests. The dielectric sphere is shown in red.
• Figure 3: Convergence of the normalized $L^2$ error of the electric potential for the dielectric sphere test case with different polynomial degrees. The results with (solid) and without (dashed) p-adaptation are compared at different polynomial degrees. In the P-adaptive case, the polynomial degree for the elements inside the dielectric sphere and not adjacent to its boundary is set to $2$.
• Figure 4: Electric potential for two plasma sheath simulations on a mesh with four elements. In the first simulation, a low polynomial degree ($N=1$) is chosen. In the second simulation, the polynomial degree is set to $N=4$ in the elements marked in green. The analytical solution is shown in black, as a dashed line.
• Figure 5: Discretized domain and boundaries used in the simulation. The values of the boundary conditions are summarized in Table \ref{['tab:ion-optic-boundaries']}.