Volume 55, Number 5, September-October 2021
|Page(s)||2293 - 2322|
|Published online||21 October 2021|
A posteriori error estimates for semilinear optimal control problems
Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile
* Corresponding author: email@example.com
Accepted: 8 July 2021
In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we devise and analyze a reliable and efficient a posteriori error estimator for a semilinear optimal control problem; control constraints are also considered. We consider a fully discrete scheme that discretizes the state and adjoint equations with piecewise linear functions and the control variable with piecewise constant functions. The devised error estimator can be decomposed as the sum of three contributions which are associated to the discretization of the state and adjoint equations and the control variable. We extend our results to a scheme that approximates the control variable with piecewise linear functions and also to a scheme that approximates the solution to a nondifferentiable optimal control problem. We illustrate the theory with two and three-dimensional numerical examples.
Mathematics Subject Classification: 35J61 / 49J20 / 49M25 / 65N15 / 65N30
Key words: optimal control problems / semilinear equations / finite element approximations / a posteriori error estimates
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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