A multigrid approach for the solution of the 2D semiconductor equations

Abstract In this paper a multigrid method for the solution of the steady semiconductor equations is presented. The discretization is made on an adaptive grid, by means of a mixed finite element method on rectangles, with the trapezoidal quadrature rule. In this way the resulting scheme reduces to the well-known Scharfetter-Gummel discretization. The grid transfer operators are selected in accordance with the discretization. The multigrid solution method is based on a collective, symmetric five-point Vanka relaxation, and—in order to admit very coarse grids—a local damping of the coarse grid correction is applied. It is shown that the convergence rate is independent of the grid size. Since nested iteration is combined with the multigrid iteration, the resulting solution method has optimal efficiency.