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This paper extends the algorithm of Benner, Heinkenschloss, Saak, andWeichelt: An inexact low-rank Newton-ADI method for large-scale algebraicRiccati equations, Applied Numerical Mathematics Vol.~108 (2016), pp.~125--142,doi:10.1016/j.apnum.2016.05.006 to Riccati equations associated with Hessenbergindex-2 Differential Algebratic Equation (DAE) systems. Such DAE systems arise,e.g., from semi-discretized, linearized (around steady state) Navier-Stokesequations. The solution of the associated Riccati equation is important, e.g.,to compute feedback laws that stabilize the Navier-Stokes equations. Challengesin the numerical solution of the Riccati equation arise from the large-scale ofthe underlying systems and the algebraic constraint in the DAE system. Thesechallenges are met by a careful extension of the inexact low-rank Newton-ADImethod to the case of DAE systems. A main ingredient in the extension to theDAE case is the projection onto the manifold described by the algebraicconstraints. In the algorithm, the equations are never explicitly projected,but the projection is only applied as needed. Numerical experience indicatesthat the algorithmic choices for the control of inexactness and line-search canhelp avoid subproblems with matrices that are only marginally stable. Theperformance of the algorithm is illustrated on a large-scale Riccati equationassociated with the stabilization of Navier-Stokes flow around a cylinder.

Efficient Solution of Large-Scale Algebraic Riccati Equations Associated with Index-2 DAEs via the Inexact Low-Rank Newton-ADI Method