A nested-grid finite-difference Poisson solver for concentrated source terms

For the numerical simulation of electric discharges, electric fields are calculated with source terms that are concentrated on a very small part of the computational domain. Therefore, a grid adaptation procedure is useful here. However, most existing fast Poisson solvers require a uniform grid. In this paper, a solver is described that combines a nested-grid approach with an existing cyclic-reduction Poisson solver. The solution process is started on a coarse grid, on the entire computational domain. New, finer grids are placed on a part of this domain, based on an error estimate. On each grid, the solution is found using the existing solver, the boundary conditions are interpolated from the underlying coarser grid. On the fine grids, even finer grids are placed, etc. The equations that describe the motion of the electric discharge are solved on a different set of nested grids, so an interpolation procedure between these two sets of nested grids is given. This procedure is used to find the input charge and to compute the output electric field. The most important new feature of the method is the error-based refinement criterion, which allows the calculation of an upper limit for the total error. Using this error bound, the accuracy of a computation can be chosen a priori. Results show that the extra error caused by the nested-grid approach can indeed be made arbitrarily small, if desired.