FWF Project P18971-N18
Optimization of technological processes plays an increasing role in science and engineering. This project deals with different types of optimal control problems governed by elliptic or parabolic partial differential equations and characterized by additional pointwise inequality constraints for control and state. Of particular interest are problems with all kinds of singularities including those due to reentrant corners and edges, nonsmooth coefficients, small parameters, and inequality constraints. The project targets two goals: First, starting from a priori error estimates, families of meshes are generated that ensure optimal approximation rates. Second, reliable posteriori error estimators are developed and used for adaptive mesh refinement. A challenge is the incorporation of pointwise inequality constraints for control and state. Both techniques can ensure efficient and reliable numerical results. With a successful strategy it is possible to calculate numerical solutions of the optimal control problems with given accuracy at low cost. While we concentrate on control problems with a linear state equation in this proposal for the first period, we plan to consider semilinear state equations in the second period.
Keywords and AMS Classification
- optimal control
- inequality constraints
- finite element discretization
- a priori error analysis
- a posteriori error estimation
- mesh refinement
AMS Subject Classification: 49K20, 49M25, 49N10, 49N60
- Rösch, A., Vexler, B., Optimal control of the Stokes equations: A priori error analysis for finite element discretization with postprocessing, SIAM Journal Numerical Analysis, 44(5):1903--1920, 2006.
- Apel, T., Rösch, A., Winkler, G., Discretization error estimates for an optimal control problem in a nonconvex domain. In: A. Bermudez de Castro et al. (eds.): Numerical Mathematics and Advanced Applications, Proceedings of ENUMATH 2005, the 6th European Conference on Numerical Mathematics and Advanced Applications, Santiago de Compostela, Spain, July 2005, 299--307, Springer, Berlin, 2006
- Apel, T., Rösch, A., Winkler, G., Optimal control in non-convex domains: a priori discretization error estimates, Calcolo 44: 137--158, 2007.
- Vexler, B., Wollner, W., Adaptive finite elements for elliptic optimization problems with control constraints, SIAM Journal on Control and Optimization, 47(1):509--534, 2008.
- Apel, T., Winkler, G., Optimal Control Under Reduced Regularity, Appl. Numer. Math., 59:2050--2064, 2009.
- Meidner, D., Vexler, B., A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems. Part I: Problems without Control Constraints, SIAM Journal on Control and Optimization, 47(3):1150--1177, 2008.
- Meidner, D., Vexler, B., A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems. Part II: Problems with Control Constraints, SIAM Journal on Control and Optimization, 47(3):1301--1329, 2008.
- Benedix, O., Vexler, B., A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints, accepted for publication in Computational Optimization and Applications, 2008.
- Apel, T., Sirch, D., L^2-error estimates for the Dirichlet and Neumann problem on anisotropic finite element meshes, submitted, 2008.
- Merino, P., Tröltzsch, F., Vexler, B., Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space, submitted, 2008.