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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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Subspace and auxiliary space preconditioners for high-order interior penalty discretizations in H (div)
In this paper, we construct and analyze preconditioners for the interior penalty discontinuous Galerkin discretization posed in the spaceH(div). These discretizations are used as one component in exactly divergence-free pressure-robust discretizations for the Stokes problem. Three preconditioners are presently considered: a subspace correction preconditioner using vertex patches and the lowest-orderH1-conforming space as a coarse space, a fictitious space preconditioner using the degree-pdiscontinuous Galerkin space, and an auxiliary space preconditioner using the degree-(p− 1) discontinuous Galerkin space and a block Jacobi smoother. On certain classes of meshes, the subspace and fictitious space preconditioners result in provably well-conditioned systems, independent of the mesh sizeh, polynomial degreep, and penalty parameterη. All three preconditioners are shown to be robust with respect tohon general meshes, and numerical results indicate that the iteration counts grow only mildly with respect topin the general case. Numerical examples illustrate the convergence properties of the preconditioners applied to structured and unstructured meshes. These solvers are used to construct block-diagonal preconditioners for the Stokes problem, which result in uniform convergence when used with MINRES.
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- PAR ID:
- 10676865
- Publisher / Repository:
- ESAIM
- Date Published:
- Journal Name:
- ESAIM: Mathematical Modelling and Numerical Analysis
- Volume:
- 59
- Issue:
- 4
- ISSN:
- 2822-7840
- Page Range / eLocation ID:
- 1909 to 1936
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1051/m2an/2025045
