A note on variational discretization of elliptic Neumann boundary control

We consider variational discretization of Neumann-type elliptic optimal control problems with constraints on the control. In this approach the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of linear finite elements. The control variable is not discretized. Optimal error bounds for control and state are obtained both in two and three space dimensions. Finally, we discuss some implementation issues of a generalized Newton method applied to the numerical solution of the problem class under consideration.

[1]  Michael Hinze,et al.  A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case , 2005, Comput. Optim. Appl..

[2]  M. Dowell,et al.  The “Pegasus” method for computing the root of an equation , 1972 .

[3]  Michael Ulbrich,et al.  Semismooth Newton Methods for Operator Equations in Function Spaces , 2002, SIAM J. Optim..

[4]  Kazufumi Ito,et al.  The Primal-Dual Active Set Strategy as a Semismooth Newton Method , 2002, SIAM J. Optim..

[5]  Fredi Tröltzsch,et al.  Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems , 2005, Comput. Optim. Appl..

[6]  C. Bernardi.  Optimal finite-element interpolation on curved domains , 1989 .

[7]  Jinchao Xu,et al.  Some Nonoverlapping Domain Decomposition Methods , 1998, SIAM Rev..

[8]  Andreas Günther,et al.  Hamburger Beiträge zur Angewandten Mathematik Finite element approximation of Dirichlet boundary control for elliptic PDEs on two-and three-dimensional curved domains , 2008 .

[9]  Boris Vexler,et al.  Finite Element Approximation of Elliptic Dirichlet Optimal Control Problems , 2007 .

[10]  Stefan Ulbrich,et al.  Optimization with PDE Constraints , 2008, Mathematical modelling.

[11]  Alfred H. Schatz,et al.  Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates , 1998, Math. Comput..

[12]  Michael Hinze,et al.  Convergence of a Finite Element Approximation to a State-Constrained Elliptic Control Problem , 2007, SIAM J. Numer. Anal..

[13]  Eduardo Casas,et al.  Error estimates for the numerical approximation of Neumann control problems , 2008, Comput. Optim. Appl..

[14]  Michael Hinze,et al.  Variational discretization and semi-smooth Newton methods; implementation, convergence and globalization in pde constrained optimization with control constraints , 2009, 0912.0488.

[15]  Jean-Pierre Raymond,et al.  ESTIMATES FOR THE NUMERICAL APPROXIMATION OF DIRICHLET BOUNDARY CONTROL FOR SEMILINEAR ELLIPTIC EQUATIONS , 2006 .