On the Applicationn of Preconditioning Operators for Nonlinear Elliptic Problems

Abstract

In this paper we study the preconditioned iterative solution of nonlinear elliptic problems

$$ \left. \begin{gathered} - div f(x,\nabla u) = g(x) in \Omega , \hfill \\ u_{|\partial \Omega } = 0 \hfill \\ \end{gathered} \right\} $$

(1)

on a bounded domain Ω ⊂ ℝ^N with suitable ellipticity conditions. Such problems arise in many applications in physics and other fields, for instance in elasto-plasticity, magnetic potential equations and subsonic flow problems [7], [15], [16]. In these problems the nonlinearity f often has the form

$$ f(x,\eta ) = a(|\eta |)\eta $$

(2)

with some positive real C ¹ function, i.e., the corresponding operator is

$$ T(u) = - div(a(|\nabla u|)\nabla u). $$