The Approximation Problem for Drift-Diffusion Systems

This review surveys a significant set of recent ideas developed in the study of nonlinear Galerkin approximation. A significant role is played by the Krasnosel’skii Calculus, which represents a generalization of the classical inf-sup linear saddle point theory. A description of a proper extension of this calculus, and the relation to the inf-sup theory are part of this review. The general study is motivated by steady-state, self-consistent, drift-diffusion systems. The mixed boundary value problem for nonlinear elliptic systems is studied with respect to defining a sequence of convergent approximations, satisfying requirements of: (1) optimal convergence rate; (2) computability; and, (3) stability. It is shown how the fixed point and numerical fixed point maps of the system, in conjunction with the Newton-Kantorovich method applied to the numerical fixed point map, permit a solution of this approximation problem. A critical aspect of the study is the identification of the breakdown of the Newton-Kantorovich method, when applied to the differential system in an approximate way. This is now known as the numerical loss of derivatives. As an antidote, a linearized variant of successive approximation, with locally defined subproblems bounded in number at each iteration, is demonstrated. In (2), a distinction is made between the outer analytical iteration, and the inner iteration, governed by numerical linear algebra. The systems studied are broad enough to include important application areas in engineering and science, for which significant computational experience is available.