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Optimal control of a free boundary problem with surface tension effects: a priori error analysis. (English) Zbl 1343.49043

The authors of the article consider the optimal control of a model free boundary problem with surface tension effects in a variational form. The state system couples the Laplace equation in the bulk with the Young-Laplace equation on the free boundary to account for surface tension. At first, the authors prove that for boundary data in \(W^2_p\), \(p>2\), the state and adjoint systems have strong solutions (the required regularity for the error analysis). Then, a fully discrete optimization problem, using piecewise linear finite elements, is introduced and error estimates in \(W^1_p\times W^1_\infty\) for the state variables and in \(W^1_q\times W^1_1\) for the adjoint variables are proved. An \(L^2\)-error bound for the optimal control is also derived. This entails using the second order sufficient optimality conditions of [H. Antil et al., SIAM J. Control Optim. 52, No. 5, 2771–2799 (2014; Zbl 1311.49008)] and first order necessary optimality conditions for both the continuous and discrete systems. Finally, two numerical examples are given. The first example explores the unconstrained problem, whereas the second example deals with the constrained one.

MSC:

49M25 Discrete approximations in optimal control
49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
35R35 Free boundary problems for PDEs
35Q93 PDEs in connection with control and optimization
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs

Citations:

Zbl 1311.49008

Software:

deal.ii; NewtonLib
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Full Text: DOI arXiv

References:

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