A multigrid solver for the semiconductor equations

Bernhard Bachmann

Institut für Angewandte Mathematik der Universität
Zürich, Rämistr.74, 8001 Zürich, Switzerland
and
Asea Brown Boveri, Corporate Research,
5405 Baden-Dättwil, Switzerland.


Abstract

We present a multigrid solver for the exponential fitting method, applied to the current continuity equations of semiconductor device simulation in two dimensions. The exponential fitting method is based on a mixed finite element discretization using the lowest-order Raviart-Thomas triangular element. This discretization method yields a good approximation of front layers and guarantees current conservation. The corresponding stiffness matrix is an M-matrix. "Standard" multigrid solvers, however, cannot be applied to the resulting system, as this is dominated by an extremely unsymmetric part, which is due to the presence of strong convection in part of the domain. To overcome this difficulty, we explore the connection between Raviart-Thomas mixed methods and the non-conforming Crouzeix-Raviart finite element discretization. In this way we can construct nonstandard prolongation and restriction operators using easily computable weighted L^2-projections, based on suitable quadrature rules and the upwind effects of the discretization. The resulting multigrid algorithm shows very good results, even for real-world problems and for locally refined grids.