Preconditioning for boundary control problems in incompressible fluid dynamics

TL;DR

This paper investigates the use of modified preconditioners to improve the efficiency of solving boundary control problems in incompressible fluid dynamics, specifically for Stokes and Navier--Stokes equations.

Contribution

It introduces a modified preconditioner tailored for Stokes boundary control and explores its potential for Navier--Stokes boundary control, filling a gap in existing solver techniques.

Findings

Modified preconditioner improves solver efficiency for Stokes boundary control.

Preconditioner shows promise for Navier--Stokes boundary control.

Numerical results support the effectiveness of the proposed approach.

Abstract

PDE-constrained optimization is a field of numerical analysis that combines the theory of PDEs, nonlinear optimization and numerical linear algebra. Optimization problems of this kind arise in many physical applications, prominently in incompressible fluid dynamics. In recent research, efficient solvers for optimization problems governed by the Stokes and Navier--Stokes equations have been developed which are mostly designed for distributed control. Our work closes a gap by showing the effectiveness of an appropriately modified preconditioner to the case of Stokes boundary control. We also discuss the applicability of an analogous preconditioner for Navier--Stokes boundary control and provide some numerical results.