Direct simulation of filamentary gas discharges like streamers or dielectric barrier micro-discharges requires the use of an adaptive mesh. The objective of this paper is to develop a strategy which can use a set of grids with suitable local refinements for the continuity equations and Poisson's equation in 2D and 3D geometries with a high-order discretization. The advantages of this approach are presented with a filamentary discharge simulation in plane–plane geometry in nitrogen within the diffusion–drift approximation.