Heiko K. Weichelt

Riccati-based Boundary Feedback Stabilization of Incompressible Navier--Stokes Flows

In this article a boundary feedback stabilization approach for incompressible Navier--Stokes flows is studied. One of the main difficulties encountered is the fact that after space discretization by a mixed finite element method (because of the solenoidal condition) one ends up with a differential algebraic system of index 2. The remedy here is to use a discrete realization of the Leray projection used by Raymond [J.-P. Raymond, SIAM J. Control Optim., 45 (2006), pp. 790--828] to analyze and stabilize the continuous problem. Using the discrete projection, a linear quadratic regulator (LQR) approach can be applied to stabilize the (discrete) linearized flow field with respect to small perturbations from a stationary trajectory. We provide a novel argument that the discrete Leray projector is nothing else but the numerical projection method proposed by Heinkenschloss and colleagues in [M. Heinkenschloss, D. C. Sorensen, and K. Sun, SIAM J. Sci. Comput., 30 (2008), pp. 1038--1063]. The nested iteration resul...

Efficient Solution of Large-Scale Saddle Point Systems Arising in Riccati-Based Boundary Feedback Stabilization of Incompressible Stokes Flow

We investigate numerical methods for solving large-scale saddle point systems which arise during the feedback control of flow problems. We focus on the instationary Stokes equations that describe instationary, incompressible flows for moderate viscosities. After a mixed finite element discretization we get a differential-algebraic system of differential index two [J. Weickert, Navier-Stokes Equations as a Differential-Algebraic System, Preprint SFB393/96-08, Department of Mathematics, Chemnitz University of Technology, Chemnitz, Germany, 1996]. To reduce this index, we follow the analytic ideas of [J.-P. Raymond, SIAM J. Control Optim., 45 (2006), pp. 790--828] coupled with the projection idea of [M. Heinkenschloss, D. C. Sorensen, and K. Sun, SIAM J. Sci. Comput., 30 (2008), pp. 1038--1063]. Avoiding this explicit projection leads to solving a series of large-scale saddle point systems. In this paper we construct iterative methods to solve such saddle point systems by deriving efficient preconditioners b...

Optimal Control-Based Feedback Stabilization of Multi-Field Flow Problems

We discuss the numerical solution of the feedback stabilization problem for multi-field flow problems. Our approach is based on an analytical Riccati feedback concept derived by Raymond which allows to steer a perturbed flow back to its desired state, assumed to be a stationary, possibly unstable, flow profile. This concept, originally derived for incompressible flow fields described by the Navier-Stokes equations, uses a linear-quadratic regulator (LQR) approach for the linearized Navier-Stokes equations formulated on the space of divergence-free velocity fields. We extend this approach to a setting where the Navier-Stokes equations are coupled to a diffusion-convection equation describing the transport of a reactive species in a fluid. The stabilizing feedback control resulting from the LQR problem is obtained via solving the associated operator Riccati equation. We describe a numerical procedure to solve this Riccati equation numerically. This involves several technical difficulties on the algebraic level that we address in this report. We illustrate the performance of our method by a numerical example.

A non-conforming composite quadrilateral finite element pair for feedback stabilization of the Stokes equations

Abstract - In this contribution, we show a method for the boundary feedback stabilization of the Stokes problem around a stationary trajectory. We derive a formal low-rank algorithm for solving the stabilization problem in operator notation. The appearing operator equations are formulated in terms of stationary partial differential equations (PDEs) instead of using their finite dimensional representations in terms of matrices. A Galerkin method, satisfying the divergence constraint pointwise locally is especially appealing since it represents appropriately the action of the Helmholtz projection. The main advantages of the composite technique are the efficient assembly of element matrices, the reduction of computational costs using static condensation, and the diagonal mass matrix. The non-conforming character of the composite element guarantees a better sparsity pattern, compared to conforming elements, due to the lower number of couplings between basis functions corresponding to neighboring cells. We also achieve the pointwise mass conservation on sub-triangles of each element.

Efficient Solvers for Large-Scale Saddle Point Systems Arising in Feedback Stabilization of Multi-field Flow Problems

This article introduces a block preconditioner to solve large-scale block structured saddle point systems using a Krylov-based method. Such saddle point systems arise, e.g., in the Riccati-based feedback stabilization approach for multi-field flow problems as discussed in [2]. Combining well known approximation methods like a least-squares commutator approach for the Navier-Stokes Schur complement, an algebraic multigrid method, and a Chebyshev-Semi-Iteration, an efficient preconditioner is derived and tested for different parameter sets by using a simplified reactor model that describes the spread concentration of a reactive species forced by an incompressible velocity field.