The first case considers a time-of-flight experiment at atmospheric pressure of 760 Torr and a constant gas temperature of 300 K. The configuration consists of two plane-parallel electrodes with a gap distance of 1 mm and a radius of 0.5 mm. The constant field of 3.55 MV/m was assumed in order to model the spatiotemporal evolution of an electron cloud.

\nThe second case considers a positive streamer in atmospheric-pressure air and a gas temperature of 300 K. Two electrodes are set 1.25 cm apart, while their radius is 1.25 cm. The constant voltage of -18.75 kV was applied at the powered electrode, while an additional Gaussian seed of ions was introduced near the anode to enhance the electric field above the breakdown threshold locally.

\nThe third case considers abnormal glow discharge in argon at low pressure of 1 Torr. The plane-parallel electrodes with the radius of 1 cm were set 1 cm apart. The exponentially rising voltage with U_0 = -250 V and time constant of 1 ns was applied to the powered electrode.

\n" }, "medium": { "name": "air, Ar", "properties": "First case (time-of-flight experiment): synthetic air, gas pressure is 760 Torr, constant gas temperature of 300 K

\nSecond case (streamer benchmark): synthetic air, gas pressure is 760 Torr, constant gas temperature of 300 K

\nThird case (abnormal glow discharge): pure argon, gas pressure is 1 Torr, constant gas temperature of 300 K

\n" }, "target": [], "diagnostics": { "name": "fluid-Poisson model", "properties": "Model: time-dependent fluid-Poisson model in spatially two-domensional (2D) cylindrical geometry;

\nComputational software: finite element discharge modelling (FEDM) code (https://github.com/INP-PM/FEDM/releases/tag/v1.0.0), powered by FEniCS (https://fenicsproject.org); COMSOL Multiphysics\u00ae v. 5.6 (www.comsol.com, COMSOL AB, Stockholm, Sweden) as a benchmark;

\nNumerical method: finite element method; element type: linear Lagrange elements for particle balance equations and Poisson equation; time-stepping: Backward differentiation formula (BDF), with automatically adapted time-step size;

\nSolver: fully coupled Newton solver, with direct linear (MUMPS for FEDM, and PARDISO and MUMPS for COMSOL Multiphysics\u00ae) and iterative (gmres with hypre preconditioner) solver; nonlinear solver tolerance: 0.0001; time stepping tolerance: varied for different case studies;

\n", "procedure": "The FEDM code verification was performed using the method of exact solutions and benchmarking. In the first study, the time-of-flight experiment is modelled. The calculated electron number density was compared to the analytically derived exact solution, and rigorous code verification was carried out by performing order-of-accuracy studies for the space and time discretisation of the particle balance equation. The modelling of the positive streamer benchmark in the air at atmospheric pressure proposed by Bagheri B et al. 2018 Plasma Sources Sci. Technol. 27 095002 was used for a further, less rigorous verification of the FEDM code. This benchmarking is useful for verifying modelling codes for cases where an analytical solution is unavailable. The modelling of an abnormal glow discharge in argon at low pressure was used to illustrate all FEDM code features and for further benchmarking by comparing the results to the commercial software package COMSOL Multiphysics\u00ae. The performance of the code was compared to the commercial software, and the influence of the choice of the linear solver on the code performance was tested by measuring the parallel speed-up.

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