A Posteriori Error Estimates for a Distributed Optimal Control Problem of the Stationary Navier--Stokes Equations

Optimal control of a stationary Navier-Stokes hemivariational inequality with numerical approximation

“…The control of fluid flow has garnered sustained attention from both engineers and scientists prompted by the demand for intricate technological applications. Optimal control problems subject to the regular boundary value problems of the Navier-Stokes equations are studied in a variety of publications, e.g., [1,4]. Optimal control problems related to variational-hemivariational inequalities have also been studied in various papers, e.g., [9,24,33].…”

A posteriori error analysis for a distributed optimal control problem governed by the von Kármán equations

“…To the best of our knowledge, there are no results in literature that discuss a posteriori error analysis for the approximation of regular solutions of optimal control problems governed by von Kármán equations. Recently, a posteriori error analysis for the optimal control problem governed by second-order stationary Navier-Stokes equations is studied in [1] using conforming finite elements under smallness assumption on the data. The trilinear form in [1] vanishes whenever the second and third variables are equal, and satisfies the anti-symmetric property with respect to the second and third variables and this aids the a posteriori error analysis.…”

“…Recently, a posteriori error analysis for the optimal control problem governed by second-order stationary Navier-Stokes equations is studied in [1] using conforming finite elements under smallness assumption on the data. The trilinear form in [1] vanishes whenever the second and third variables are equal, and satisfies the anti-symmetric property with respect to the second and third variables and this aids the a posteriori error analysis. This paper discusses approximation of regular solutions for fourth-order semilinear problems without any smallness assumption on the data.…”

“…They have lesser number of degrees of freedom in comparison with the conforming Argyris finite elements with 21 degrees of freedom in a triangle or the Bogner-Fox-Schmit finite elements with 16 degrees of freedom in a rectangle. The discrete trilinear form 𝑏 NC (•, •, •) := 1 The a posteriori analysis for the fully discrete optimal control problem governed by the von Kármán equations addressed in this article is novel. For example, this formulation is different from that in [19], and it is essential to modify the a priori error estimates for the discrete optimal problem.…”

“…for all 𝑣 ∈ C ū, where Θ = Θ(ū) is the associated adjoint state and z𝑣 = 𝑧 𝑣 (ū) is the solution to (3.2) for 𝑢 = ū and 𝑣 ∈ C ū. Theorem 3.9 (Second-order sufficient condition [1,16]). Let (︀ Ψ, ū)︀ be a nonsingular local solution of (2.1) and let Θ = Θ(ū) be the associated adjoint state.…”