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Abstract We propose a uniform block-diagonal preconditioner for condensed $$H$$(div)-conforming hybridizable discontinuous Galerkin schemes for parameter-dependent saddle point problems, including the generalized Stokes equations and the linear elasticity equations. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning technique (Xu (1994) The Auxiliary Space Method and Optimal Multigrid Preconditioning Techniques for Unstructured Grids, vol. 56. International GAMM-Workshop on Multi-level Methods (Meisdorf), pp. 215–235). A spectrally equivalent approximation to the Schur complement on the element-wise constant pressure space is also constructed, and an explicit computable exact inverse is obtained via the Woodbury matrix identity. Finally, the numerical results verify the robustness of our proposed preconditioner with respect to model parameters and mesh size.
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This content will become publicly available on October 22, 2026
Preconditioning of the generalized Stokes problem arising from the approximation of the time-dependent Navier-Stokes equations
The paper compares standard iterative methods for solving the generalized Stokes problem arising from the time and space approximation of the time-dependent incompressible Navier-Stokes equations. Various preconditioning techniques are considered: (1) pressure Schur complement; (2) fully coupled system using an exact factorization as a basis for the preconditioner; (3) fully coupled system using norm equivalence considerations as a basis for the preconditioner; (4) in all the cases we also investigate the benefits of the augmented Lagrangian formulation. Our objective is to see whether one of these methods can compete with traditional pressure-correction and velocity-correction methods in terms of throughput (the throughput is the ratio of the number of degrees of freedom of the problem divided by the number of cores and the wall-clock time in second). Numerical tests on fine unstructured meshes (68 millions degrees of freedoms) demonstrate GMRES/CG convergence rates that are independent of the mesh size and improve with the Reynolds number for most methods. Three conclusions are drawn: (1) The throughputs of all the methods tested in the paper are similar up to mesh-independent multiplicative constants (see Fig. 6). (2) Although very good parallel scalability is observed for the augmented Lagrangian version of the generalized Stokes problem, the best throughputs are achieved without the augmented Lagrangian term. (3) The throughput of all the methods tested in the paper is on average 5 to 25 times slower than that of traditional pressure-correction and velocity-correction methods (on average 5 for the best one). Hence, although all these methods are very efficient for solving steady state problems, pressure-
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- Award ID(s):
- 2110868
- PAR ID:
- 10640475
- Publisher / Repository:
- 2025 Elsevier Ltd., Computers and Mathematics with Applications 191 (2025) 255–274
- Date Published:
- Journal Name:
- Computers and Mathematics with Applications
- ISSN:
- 1873-7668
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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