Solution of equations: the multigrid method

Abstract

The equations governing the modelling of many semiconductor devices, including the HEMT, are differential equations with associated boundary conditions. These equations are discretised on a grid which is fine enough to preserve all of the necessary physical detail. In an iterative solution of the equations on this fine grid, the high frequency components of the errors are rapidly eliminated, but the convergence then becomes slow. In the multigrid method, the equations and their partial solutions are moved up and down through a succession of grids, from the initial fine grid to a very coarse grid, and different error frequencies are eliminated on different grids. This method allows the equations to be solved on a fine grid with a significant decrease in computing time. An introduction to the multigrid method is given, and some of the specific problems which arise in the application of this method to device modelling are discussed.