Adaptive multigrid applied to a bipolar transistor problem Original Research Article

In this paper an adaptive nonlinear multigrid method is presented for the solution of the steady 2D semiconductor equations. The discretization is made on an adaptive grid by means of the (hybrid) mixed finite element method on rectangles. The integrals involved are approximated by means of the trapezoidal rule in order to obtain a generalization in 2D of the well-known Scharfetter-Gummel scheme. We show that the use of the trapezoidal rule does not influence the accuracy of the discretization. The discrete equations thus obtained are solved by means of the dual version of the FAS-FMG algorithm. A Vanka-type relaxation is used as a smoother, and a local damping of the restricted residual is introduced in order to be able to use very coarse grids. Consistent with the FAS-FMG algorithm, we use the relative truncation error between coarse and fine grids as a refinement criterion for constructing adaptive grids. We study the relative truncation errors for the semiconductor equations in detail and show how they can be incorporated into a practical grid adaptation scheme. Results are shown for a realistic bipolar transistor problem.