This review surveys a significant set of recent ideas developed in the
study of nonlinear Galerkin approximation. A significant role is play
ed by the Krasnosel'skii calculus, which represents a generalization o
f the classical inf-sup linear saddlepoint theory. A description of a
proper extension of this calculus and the relation to the inf-sup theo
ry are part of this review. The general study is motivated by steady-s
tate, self-consistent, drift-diffusion systems. The mixed boundary val
ue problem for nonlinear elliptic systems is studied with respect to d
efining a sequence of convergent approximations, satisfying requiremen
ts of: (i) optimal convergence rate; (ii) computability; and, (iii) st
ability. It is shown how the fixed point and numerical fixed point map
s of the system, in conjunction with the Newton-Kantorovich method app
lied to the numerical fixed point map, permit a solution of this appro
ximation problem. A critical aspect of the study is the identification
of the breakdown of the Newton-Kantorovich method, when applied to th
e 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 (ii), a distinction is
made between the outer analytical iteration, and the inner iteration,
governed by numerical linear algebra. The systems studied are broad en
ough to include important application areas in engineering and science
, for which significant computational experience is available.