Error Estimates for the Numerical Approximation of a Distributed Control Problem for the Steady-State Navier–Stokes Equations

Abstract

We obtain error estimates for the numerical approximation of a distributed control problem governed by the stationary Navier-Stokes equations, with pointwise control constraints. We show that the L 2-norm of the error for the control is of order h 2 if the control set is not discretized, while it is of order h if it is discretized by piecewise constant functions. These error estimates are obtained for local solutions of the control problem, which are nonsingular in the sense that the linearized Navier-Stokes equations around these solutions define some isomorphisms, and which satisfy a second order sufficient optimality condition. We establish a second order necessary optimality condition. The gap between the necessary and sufficient second order optimality conditions is the usual gap known for finite dimensional optimization problems.

Error estimates for the discretization of the velocity tracking problem

“…Therefore, it is not surprising that the discretization parameter τ is needed to be small compared with h if we want to prove the uniqueness of the solution for the fully discrete system. The key idea of [6] was to utilize ideas from [8] developed for the stationary Navier-Stokes, together with a detailed error analysis of the uncontrolled state and adjoint equations of the underlying scheme.…”

“…Here, we emphasize that the convective nature of the adjoint equation also requires special attention. Then, we combine these estimates within the framework of [10,8,7] (related to nonlinear elliptic pde control constrained problems), by exploring a localization argument and the second order condition. To our best knowledge our estimates are new.…”

Optimal Boundary Control Problems Related to High-Lift Configurations

“…If g is small compared to ν, then the solution u is unique. The assumption of non-singularity implies that the Navier-Stokes equation is uniquely solvable in a neighborhood of the reference control and state, we refer to [7,Theorem 2.5] Notice that our optimal control problem is nonlinear and hence belongs to the field of nonconvex optimization. Therefore, we have to deal with locally optimal solutions.…”

Finite Element Discretization in Shape Optimization Problems for the Stationary Navier-Stokes Equation

“…According, for instance, to Temam [1979], there are many possibilities to introduce a finite element space V h in Ω h approximating (2), that is approximating H 1 0 (Ω) and the divergence free condition. In particular, the piecewise linear finite elements are not possible to be used in this setting.…”