Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Hafemeyer Dominik; Mannel Florian; Neitzel Ira; Vexler Boris

doi:10.3934/mcrf.2019041

Article Contents

2020, Volume 10, Issue 2: 333-363. Doi: 10.3934/mcrf.2019041

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Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

1.
Department of Mathematics, Technische Universität München, Boltzmannstr. 3, 85748 Garching b. München, Germany
2.
Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstr. 36, 8010 Graz, Austria
3.
Institute for Numerical Simulation, Universität Bonn, Endenicher Allee 19b, 53115 Bonn, Germany

^* Corresponding author: Dominik Hafemeyer
^* Corresponding author: Dominik Hafemeyer

Received: January 31, 2019

Revised: May 31, 2019

Early access: August 2019

Published: April 2020

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Abstract

We consider an optimal control problem governed by a one-dimensional elliptic equation that involves univariate functions of bounded variation as controls. For the discretization of the state equation we use linear finite elements and for the control discretization we analyze two strategies. First, we use variational discretization of the control and show that the $ L^2 $- and $ L^\infty $-error for the state and the adjoint state are of order $ {\mathcal O}(h^2) $ and that the $ L^1 $-error of the control behaves like $ {\mathcal O}(h^2) $, too. These results rely on a structural assumption that implies that the optimal control of the original problem is piecewise constant and that the adjoint state has nonvanishing first derivative at the jump points of the control. If, second, piecewise constant control discretization is used, we obtain $ L^2 $-error estimates of order $ \mathcal{O}(h) $ for the state and $ W^{1, \infty} $-error estimates of order $ \mathcal{O}(h) $ for the adjoint state. Under the same structural assumption as before we derive an $ L^1 $-error estimate of order $ \mathcal{O}(h) $ for the control. We discuss optimization algorithms and provide numerical results for both discretization schemes indicating that the error estimates are optimal.

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Mathematics Subject Classification: 26A45, 49J20, 49M25, 65N15, 65N30.

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Figure 1. Example 1: A semi-discrete solution to the data from Section 5.3. The discretization parameter $ h $ is roughly $ 3.8\cdot 10^{-6} $. The inclusions provided in Corollary 2 are clearly visible

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Figure 2. Example 1: Convergence plots of the errors of the solutions to the semi-discrete problem (P_vd) compared to the exact solution. The exact solution is known

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Figure 3. Example 1: Convergence plots of the errors of the solutions to the fully discrete problem (P_cd) compared to the exact solution. The exact solution is known

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Figure 4. Example 2: A variationally discrete solution to the data from Section 5.4. The discretization parameter $ h $ is roughly $ 3.8\cdot 10^{-6} $. The inclusions provided in Corollary 2 are clearly visible

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Figure 5. Example 2: Convergence plots of the errors of the solutions to the semi-discrete problem (P_vd) compared to an approximation of the exact solution. The reference solution is computed as solution to (P_vd) with $ h_{\text{ref}}\approx 3.8\cdot 10^{-6} $

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Figure 6. Example 2: Convergence plots of the errors of the solutions to the fully discrete problem (P_cd) compared to an approximation of the exact solution. The reference solution is computed as solution to (P_cd) with $ h_{\text{ref}}\approx 2.4\cdot 10^{-7} $

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