Additive Schwarz solvers for <em>hp</em>-FEM discretizations of PDE-constrained optimzation problems

Abstract

In this paper, we investigate the minimization of a quadratic functional subject to a boundary value problem of a second order linear elliptic partial differential equation. There are no inequality constraints for state and control. This problem is discretized by <em>hp</em>-finite elements. The main focus of this talk is the development of efficient solution methods for the corresponding system of linear algebraic equations. From the literature it is known that this system is symmetric and indefinite. This paper considers two different iterative solvers<br /> • a conjugate gradient (CG) method in a special inner product, following Schöberl/Zulehner, and<br />• the minimal residual method.<br /> In both methods, efficient preconditioners for finite element mass and stiffness matrix acclerate the convergence speed of the underlying iterative method. This contribution presents overlapping <em>hp</em>- FEM preconditioners for mass and stiffness matrix. Optimal condition number estimates are proved. Robustness with respect to the regularization parameter can be shown for the CG-method. Finally several numerical examples show the efficiency of the presented algorithm.