A Multigrid Preconditioner for the Semiconductor Equations - Meza, Juan C.; Tuminaro, Ray S. (SIAM journal on scientific computing, 1996)
A multigrid preconditioned conjugate gradient algorithm is introduced into a semiconductor device modeling code DANCIR. This code simulates a wide variety of semiconductor devices by numerically solving the drift-diffusion equations. The most time-consuming aspect of the simulation is the solution of three linear systems within each iteration of the Gummel method. The original version of DANCIR uses a conjugate gradient iteration preconditioned by an incomplete Cholesky factorization. In this paper, we consider the replacement of the Cholesky preconditioner by a multigrid preconditioner. To adapt the multigrid method to the drift-diffusion equations, interpolation, projection, and coarse grid discretization operators need to be developed. These operators must take into account a number of physical aspects that are present in typical devices: wide-scale variation in the partial differential equation (PDE) coefficients, small-scale phenomena such as contact points, and an oxide layer. Additionally, suitable relaxation procedures must be designed that give good smoothing numbers in the presence of anisotropic behavior. The resulting method is compared with the Cholesky preconditioner on a variety of devices in terms of iterations, storage, and run time.