Two-sided estimates are derived for the approximation of solutions to the driftdiffusion steady-state semiconductor device system which are identified with fixed points of Gummel's solution map. The approximations are defined in terms of fixed points of numerical finite element discretization maps. By use of a calculus developed by Krasnosel'skii and his coworkers, it is possible both to locate approximations near fixed points in an "a priori" manner, as well as fixed points near approximations in an '% posteriori" manner. These results thus establish a nonlinear approximation theory, in the energy norm, with rate keyed to what is possible in a standard linear theory. This analysis provides a convergence theory for typical computational approaches in current use for semiconductor simulation.1. Introduction. The drift-diffusion model of a steady-state semiconductor device is formed by a system of three coupled partial differential equations (PDEs).This system of PDEs is solved by a solution vector of three function components. A fixed point mapping T :x Tx can be defined by solving each of these PDEs for its corresponding component and substituting these components in successive PDEs in a Gauss-Seidel fashion. Fixed points of such a mapping T then coincide with solutions to the drift diffusion model. Iteration with the mapping T defines an algorithm for the solution of the drift-diffusion model in which the PDEs are decoupled. The mapping T, termed Gummel's map [5] in the literature, is defined through solution for the potential u, for given electron and hole quasi-Fermi levels v and w, as a fractional step, and subsequently through solution for the electron and hole quasi-Fermi levels. This definition specifies the range of the mapping T. Principal properties, including fixed points and maximum principles, have been studied in increasing generality in [21],[23], and [8]. For this mapping, and for the slightly different mapping that operates in the space of the Slotboom variables V e -v and W e , the Lipschitz constant L T has been examined in detail in [15],[9],[17],[16]. Employing either quasi-Fermi levels or Slotboom variables, the appropriate formulation for device applications is the mixed Dirichlet/Neumann boundary value problem, taken over the physical device ( assumed to be a polyhedral domain, with possible solution gradient singularities at boundary transition points [15],[9].A companion approximation map is induced by piecewise linear finite elements, if the convex minimization, inherent in defining the successive gradient equations, is taken over finite dimensional affine subspaces. The fixed points of the companion map are clearly candidates for approximation of the fixed points of the solution map for the original system of PDEs. In this paper, we deduce an approximation theory, described by two-sided estimates, for this discretization procedure. Our theory is based upon an operator calculus developed by Krasnosel'skii and his collaborators (cf. [20]) in which

We present a new algorithm for the numerical simulation of electrons in a quantum wire as described by a two-dimensional eigenvalue problem for Schrödinger’s equation coupled with Poisson’s equation. Initially, the algorithm employs an underrelaxed fixed point iteration to generate an approximation which is reasonably close to the solution. Subsequently, this approximate solution is employed as an initial guess for a Jacobian-free implementation of an approximate Newton method. In this manner the nonlinearity in the model is dealt with effectively. We demonstrate the effectiveness of our approach in a set of numerical experiments which study the electron states on the cross section of a quantum wire structure based on III-V semiconductors at 4.2 and 77 K.

Summary.We examine the Lo~ stability of piecewise linear finite element approximations U to the solution u to elliptic gradient equations of the form (x, u)=g(x) where f is monotonically increasing in u. We identify a priori Loo bounds for the finite element solution U, which we call "reduced" bounds, and which are marginally weaker than those for the original differential equation. For the general, N-dimensional, case we identify new conditions on the mesh, such that under the assumption that f is Lipschitz continuous on a finite interval, U satisfies the "reduced" Lo~ bounds mentioned above. The new, N-dimensional regularity conditions preclude quasi-rectangular meshes.Moreover, we show that U is stable in Loo in two dimensions for a discretization mesh on which -V. [a(x) IZu] gives rise to an M-matrix, while Uis stable for any mesh in one dimension. The condition that the discretization ofVu] has to be an M-matrix, still allows the inclusion of the important case of triangulating in a quasi-rectangular fashion. The results are valid for either the pure Neumann problem or the general mixed Dirichlet-Neumann boundary value problem, while interfaces may be present. The boundary conditions for U are obtained by use of (nonexpansive) pointwise projection operators.

We compare both numerically and theoretically three techniques for accelerating the convergence of a nonlinear fixed point iteration u ~ T(u), arising from a coupled elliptic system: Chebyshev acceleration, a second order stationary method, and a nonlinear version of the Generalized Minimal Residual Algorithm (GMRES) which we call NLGMR. All three approaches are implemented in 'Jacobian-free' mode, i.e., only a subroutine which returns T(u) as a function of u is required.We present a set of numerical comparisons for the drift-diffusion semiconductor model. For the mapping T which corresponds to the nonlinear block GauB-Seidel algorithm for the solution of this nonlinear elliptic system, NLGMR is found to be superior to the second order stationary method and the Chebychev acceleration. We analyze the local convergence of the nonlinear iterations in terms of the spectrum a[T.(u~*~)] of the derivative Tu at the solution u ~*). The convergence of the original iteration is governed by the spectral radius p [Tv(ut*))].In contrast, the convergence of the two second order accelerations are related to the convex hull of a[Tu(ut*))], while the convergence of the GMRESbased approach is related to the local clustering in a [I-T~(ut*))]. The spectrum a [I-T.(ut*))] clusters only at 1 due to the successive inversions of elliptic partial differential equations in T. We explain the observed superiority of GMRES over the second order acceleration by its ability to take advantage of this clustering feature, which is shared by similar coupled nonlinear elliptic systems. Subject Classification (1991): 65B, 65H 10, 65N Mathematics

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