When studying the dynamics of incompressible fluids in bounded domains the only available data often provide average flow rate conditions on portions of the domain's boundary. In engineering applications a common practice to complete these conditions is to prescribe a Dirichlet condition by assuming a-priori a spatial profile for the velocity field. However, this strongly influence the accuracy of the numerical solution. A more mathematically sound approach is to prescribe the flow rate conditions using Lagrange multipliers, resulting in an augmented weak formulation of the Navier-Stokes problem. In this paper we start from the SIMPLE preconditioner, introduced so far for the standard Navier-Stokes equations, and we derive two preconditioners for the monolithic solution of the augmented problem. This can be useful in complex applications where splitting the computation of the velocity/pressure and Lagrange multipliers numerical solutions can be very expensive. In particular, we investigate the numerical performance of the preconditioners in both idealized and real-life scenarios. Finally, we highlight the advantages of treating flow rate conditions with a Lagrange multipliers approach instead of prescribing a Dirichlet condition.
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