Augmented Lagrangian acceleration of global-in-time Pressure Schur complement solvers for incompressible Oseen equations
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Date
2023-12
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Abstract
This work is focused on an accelerated global-in-time solution strategy for the Oseen
equations, which highly exploits the augmented Lagrangian methodology to improve the
convergence behavior of the Schur complement iteration. The main idea of the solution
strategy is to block the individual linear systems of equations at each time step into a
single all-at-once saddle point problem. By elimination of all velocity unknowns, the
resulting implicitly defined equation can then be solved using a global-in-time pressure
Schur complement (PSC) iteration. To accelerate the convergence behavior of this
iterative scheme, the augmented Lagrangian approach is exploited by modifying the
momentum equation for all time steps in a strongly consistent manner. While the
introduced discrete grad-div stabilization does not modify the solution of the discretized
Oseen equations, the quality of customized PSC preconditioners drastically improves
and, hence, guarantees a rapid convergence. This strategy comes at the cost that the
involved auxiliary problem for the velocity field becomes ill conditioned so that standard
iterative solution strategies are no longer efficient. Therefore, a highly specialized
multigrid solver based on modified intergrid transfer operators and an additive block
preconditioner is extended to solution of the all-at-once problem. The potential of
the proposed overall solution strategy is discussed in several numerical studies as they
occur in commonly used linearization techniques for the incompressible Navier-Stokes
equations.
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Keywords
Oseen equations, waveform relaxation, augmented Lagrangian, pressure Schur complement, parallel-in-time