The extrapolation of results obtained on a series of 3 succeeding grids with halved mesh size is tested as a variant of the multigrid approach for solving the Laplace and Poisson equations in 2D. Based on corresponding experience with BEM for electric  and magnetic  field problems a pure power law is applied instead of the famous Richardson extrapolation . On those grid points, which are common to all 3 grids, the potential values are extrapolated to an arbitrary fine discretization. On the points of the finest grid in between those of the coarser ones the potentials then are obtained by only few iterations to perform the interpolation. Both, the common 5-point discretization and the famous 9-point discretization by E. Kasper  are investigated and compared with respect to the possible win of accuracy by extrapolation. As an interesting result of this kind of extrapolation, the accumulated local discretization errors of the 5-point discretization are partially cured and the high accuracy by the 9-point formula of Kasper makes extrapolation inefficient. Like for classical MG (multi grid)  the acceleration of potential calculations on grids of large size is substantial. © 2008 Elsevier B.V. All rights reserved.
Becker, R. (2008). Improvement of FDM by extrapolation on multiple grids. In Physics Procedia (Vol. 1, pp. 245–248). https://doi.org/10.1016/j.phpro.2008.07.102