Augmented Lagrangian Acceleration of Global-in-Time Pressure Schur Complement Solvers for Incompressible Oseen Equations

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...

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Bibliographic Details
Published in:Journal of mathematical fluid mechanics Vol. 26; no. 2; p. 27
Main Authors: Lohmann, Christoph, Turek, Stefan
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.05.2024
Springer Nature B.V
Subjects:
Algorithms
Approximation
Classical and Continuum Physics
Complement
Convergence
Fluid- and Aerodynamics
Iterative solution
Linear systems
Mathematical Methods in Physics
Ordinary differential equations
Partial differential equations
Physics
Physics and Astronomy
Reynolds number
Saddle points
Solvers
Velocity
Velocity distribution
Viscosity
Oseen equations
Waveform relaxation
Pressure Schur complement
Augmented Lagrangian
Parallel-in-time
ISSN:1422-6928, 1422-6952
Online Access:Get full text
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Summary: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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ISSN:1422-6928
1422-6952
DOI:10.1007/s00021-024-00862-7