Numerical Fixed Point Approximation in Banach Space

As we have seen in the preceding chapters, the drift-diffusion model of a steady-state semiconductor device is formed by a system of three coupled partial differential equations (PDEs) for which discrete and continuous maximum principles exist. This system of PDEs is solved by a solution vector of three function components. Moreover, a fixed point mapping T can be defined. Although the definition of T is not unique, and various decouplings are possible, as was rigorously analyzed in Chap. 4, it is possible to achieve complete decoupling, via gradient equations, when the recombination term satisfies monotonicity properties, or is taken to be zero. This is carried out by solving each of these PDEs for its corresponding component, and substituting these components in successive PDEs in a Gauss-Seidel iterative fashion. Fixed points of such a mapping then coincide with solutions to the drift-diffusion model. Iteration with this mapping T defines an algorithm for the solution of the drift-diffusion model, typically termed Gummel iteration in the literature. It is really Picard iteration for the map T. The Lipschitz constant has been examined in detail in Chap. 4.