The Approximation Problem for Drift-Diffusion Systems
By: Joseph W. Jerome
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
This paper appeared in SIAM Review 37 (1995), 552--572.
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