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arXiv · 2609.31987

Finite element error estimates for a bilinear optimal control problem with pointwise tracking governed by a semilinear elliptic PDE

Enrique Otarola·Daniel Quero·Matias Sasso
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Abstract

We study finite element approximations for an optimal control problem with pointwise tracking governed by a semilinear elliptic PDE. The control variable enters the state equation as a reaction coefficient, resulting in a bilinear control, and the cost functional includes point evaluations of the state variable. We propose two discretization schemes: a semidiscrete scheme, where the control variable is not discretized, and a fully discrete scheme, where the control variable is discretized using piecewise constant functions. In two and three dimensions, for both schemes, we establish convergence of discrete solutions and prove a priori error bounds that behave as $\mathcal{O}(h|\log h|)$ in the $L^2(Ω)$-norm for approximating a locally optimal control. In two dimensions, these bounds are improved to $\mathcal{O}(h)$ for the fully discrete scheme and $\mathcal{O}(h^{2}|\log h|^2)$ for the semidiscrete scheme. We conclude with numerical experiments that demonstrate the performance of the proposed schemes.

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Enrique Otarola, Daniel Quero, Matias Sasso. 2026-09-25. Finite element error estimates for a bilinear optimal control problem with pointwise tracking governed by a semilinear elliptic PDE. https://arxiv.org/abs/2609.31987

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