Parallel Multigrid Methods for the Continuity Equations in Semiconductor Device Simulation

A Schur complement based algebraic multigrid method is discussed in view of the special conditions of the continuity equations for electrons and holes in semiconductors. The so called Scharfetter-Gummel discretization, the practically arbitrary large jumps in the solutions and the structure of the preconditioners impose special problems compared with standard finite element discretizations. The final goal is the application of the method to large three-dimensional problems - in other words: parallelization properties are an issue. The present techniques used in device simulation can not guarantee a close relation of the grids used and the coefficients of the equations. Hence a correction technique for the basis transformation is introduced. The algorithm is implemented on parallel Cray vector machines and on RISC SMPs. The performance for two- and three-dimensional examples is given.