A novel spectral element method for modelling streamer discharges and its comparison with the conventional finite-element method.

Simulating streamers in atmospheric pressure plasmas is a challenging problem due to the multiscale nature of this discharge type. The need to simulate streamers in complex geometries and the interaction of streamers with plasma-facing surfaces motivates the search for new numerical approaches to this problem. In particular, it is important to implement efficient methods for elliptic problems, which have to be dealt with in streamer simulations. A promising approach is to use a hierarchical Poincare-Steklov (HPS) scheme for solving the resulting elliptic problems [1]. The HPS scheme belongs to the family of composite spectral collocation methods and has a number of attractive features. As a proof of concept, the application of the HPS scheme to the simulation of streamer discharges has been demonstrated in [2] for the case of regular grids. In this contribution, we continue this work and compare the proposed HPS scheme with the conventional finite-element method (implemented within the FEniCS framework) in the context of streamer simulations. A suite of test problems is considered and the performance of both methods is assessed in terms of the required computational time and the number of discrete unknowns being involved. [1] P. G. Martinsson, J. Comput. Phys. 242, 460 (2013). [2] I. L. Semenov, K.-D. Weltmann, arXiv:2111.06210.