Robust finite element solvers for distributed hyperbolic optimal control problems
Abstract
We propose, analyze, and test new robust iterative solvers for systems of linear algebraic equations arising from the spacetime finite element discretization of reduced optimality systems defining the approximate solution of hyperbolic distributed, trackingtype optimal control problems with both the standard $L^2$ and the more general energy regularizations. In contrast to the usual timestepping approach, we discretize the optimality system by spacetime continuous piecewiselinear finite element basis functions which are defined on fully unstructured simplicial meshes. If we aim at the asymptotically best approximation of the given desired state $y_d$ by the computed finite element state $y_{\varrho h}$, then the optimal choice of the regularization parameter $\varrho$ is linked to the spacetime finite element meshsize $h$ by the relations $\varrho=h^4$ and $\varrho=h^2$ for the $L^2$ and the energy regularization, respectively. For this setting, we can construct robust (parallel) iterative solvers for the reduced finite element optimality systems. These results can be generalized to variable regularization parameters adapted to the local behavior of the meshsize that can heavily change in the case of adaptive mesh refinements. The numerical results illustrate the theoretical findings firmly.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.03756
 arXiv:
 arXiv:2404.03756
 Bibcode:
 2024arXiv240403756L
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Optimization and Control;
 49J20;
 49M05;
 35L05;
 65M60;
 65M15;
 65N22