Multigrid for Semiconductor Device Simulation: Cell-Centered or Vertex-CenteredMultigrid

摘要

The study of primal and dual mixed finite element discretization on rectangulargrids for the semicondutor device equations is reported. When suitable quadrature rules are used, the primal version is equivalent to the usual vertex centered box scheme, whereas the dual version leads to a cell centered finite volume discretization. In both cases the system of nonlinear equations obtained after discretization is solved for multigrid. A cell centered multigrid was used for the cell centered scheme and a vertex centered multigrid was used for the vertex centered scheme. In cell centered multigrid it is necessary to apply a local damping of the restricted residual in order to deal with the strong nonlinearity of the problem. This can be avoided by using the vertex centered multigrid algorithm provided that injection is used for the restriction of the residual. Injection is usually too inaccurate a grid transfer operator for second order differential equations, but by means of a two grid analysis it was shown that the choice of a suitable smoothing operator can alleviate this problem. In numerical experiments vertex centered multigrid appears to be more efficient and robust for solving semiconductor equations than cell centered multigrid.