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A mesh-independence result for semismooth Newton methods. (English) Zbl 1079.65065
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 Fréchet 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.
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 \(h\leq h'\), 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 \(\max\{\| y^0_h-\overline y_h\|_{L^2}\), \(\| y^0-\overline y \|_{L^2}\}\leq\delta\).
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.

MSC:
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
49J40 Variational inequalities
49K20 Optimality conditions for problems involving partial differential equations
49M25 Discrete approximations in optimal control
47J25 Iterative procedures involving nonlinear operators
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