Efficient solution of large-scale saddle point systems arising in Riccati-based boundary feedback stabilization of incompressible Stokes flow (Q2870674)

The paper focuses on the numerical treatment of the Leray projection, which is the key to a robust implementation of optimal control for flow problems. The authors investigate the non-stationary incompressible Stokes equations for moderate viscosities: NEWLINE\[NEWLINE\left.\begin{aligned}\frac{\partial}{\partial t}v(t,x)-\nu\Delta v(t,x)+\nabla p(t,x)=0\\ \nabla\cdot v(t,x)=0\end{aligned}\right\}\mathrm{ on } (0,\infty)\times\Omega\tag{1}NEWLINE\]NEWLINE with the time \(t\in(0,\infty)\), the spatial variable \(x\in\Omega\), the velocity field \(v(t,x)\in\mathbb R^2\), the pressure \(p(t,x)\in \mathbb R\), the viscosity \(\nu\in\mathbb R^+\) and \(\Omega\) being a two-dimensional bounded connected domain with boundary \(\Gamma=\partial \Omega\). Additionally, some initial and Dirichlet boundary conditions are imposed describing the inflow-outflow problem.NEWLINENEWLINEAfter the first introductory part, the second part starts by applying the method of lines to the problem (1), which discretizes the Stokes equations by a mixed finite element method in space and reduces the problem to a system of differential-algebraic equations (DAEs) of differential index two. In the sequel, in order to avoid that the solution of DAEs lies on a (hidden) manifold of the Euclidean space that implies some difficulties to the solvability, the authors use an index reduction for the numerical Leray projector. Some large-scale saddle point systems are employed for computing the Riccati feedback via this projection approach.NEWLINENEWLINEIn the third part the authors discuss the properties of the saddle point systems and develop an efficient solution strategy based on iterative methods.NEWLINENEWLINEThe fourth part describes numerical experiments for the von Kármán vortex street, showing the performance of the presented preconditioned iterative saddle point solver.