A mesh-independence result for semismooth Newton methods.

*(English)*Zbl 1079.65065The 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.

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.

Reviewer: Otu Vaarmann (Tallinn)

##### 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 |