@article{1079.65065,
author="Hinterm\"uller, Michael and Ulbrich, Michael",
title="{A mesh-independence result for semismooth Newton methods.}",
language="English",
year="2004",
doi={10.1007/s10107-004-0540-9},
abstract="{The problem is to study local properties of Newton type methods
applied to discretizations of nonsmooth operator equations $$G(y)=0,\ G:
L^2(\Omega)\to L^2(\Omega).\tag 1$$ Here the operator is related to an
MCP-function based reformulation of the infinite dimensional box-constrained
variational inequality problem. It is well known that if $G:Y\to Z$ $(Y,Z$
Banach spaces) is Fr\'echet differentiable, $G'$ is locally Lipschitz and
$G'(y)$ is invertible at a solution $\overline y$ of (1), then the Newton
method is locally quadratically convergent to $\overline y$. For approximate
discretizations: $G_h(y_h)=0$, with $G_h:Y_h\to Z_h$ and $Y_h,Z_h$ suitable
finite dimensional, the discrete Newton process possesses the property of
mesh independence, i.e. the continuous and the discrete Newton process
converge with the same rate.\par For a class of semismooth operator
equations a mesh independent result for generalized Newton methods is
established. The main result of this paper states that for given $q$-linear
rate of convergence $\theta$ there exists a sufficiently small mesh size
$h'>0$ of discretization and radius $\delta>0$ such that, for all $h\le h'$,
the continuous and the discrete Newton process converge at least at the
$q$-linear rate $\theta$ when initialized by $y^0,y^0_h$ satisfying
$\max\{\Vert y^0_h-\overline y_h\Vert_{L^2}$, $\Vert y^0-\overline y
\Vert_{L^2}\}\le\delta$.\par The mesh independent result is applied to
control a constrained control problem for semilinear elliptic partial
differential equations, for which a numerical validation of the theoretical
results are given.}",
reviewer="{Otu Vaarmann (Tallinn)}",
keywords="{Banach spaces; mesh independence; $q$-linear rate of convergence;
box-constrained variational inequality problem; control problem; semilinear
elliptic partial differential equations; nonsmooth operator equation}",
classmath="{*65J15 (Equations with nonlinear operators (numerical methods))
49J40 (Variational methods including variational inequalities)
49K20 (Optimal control problems with PDE (nec./ suff.))
49M25 (Finite difference methods)
47J25 (Methods for solving nonlinear operator equations (general))
}",
}