Numerical Verification of PolyDG Algebraic Solvers for the Pseudo-Stress Stokes Problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911417398984704 |
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| author | Antonietti, Paola F. Cancrini, Alessandra Ciaramella, Gabriele |
| author_facet | Antonietti, Paola F. Cancrini, Alessandra Ciaramella, Gabriele |
| contents | This work focuses on the development of efficient solvers for the pseudo-stress formulation of the unsteady Stokes problem, discretised by means of a discontinuous Galerkin method on polytopal grids (PolyDG). The introduction of the pseudo-stress variable is motivated by the growing interest in non-Newtonian flow models and coupled interface problems, where the stress field plays a fundamental role in the physical description. The space-time discretisation of the problem is obtained by combining the PolyDG approach in space with the implicit Euler method for time integration. The resulting linear system, characterised by a symmetric, positive, definite matrix, exhibits deteriorating convergence with standard solvers as the time step decreases. To address this issue, we investigate two tailored strategies: deflated Conjugate Gradient, which mitigates the effect of the most problematic eigenmodes, and collective Block-Jacobi, which exploits the block structure of the system matrix. Numerical experiments show that both approaches yield iteration counts effectively independent of $Δt$, ensuring robust performance with respect to the time step. Future work will focus on extending this robustness to the spatial discretisation parameter $h$ by integrating multigrid strategies with the time-robust solvers developed in this study. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02537 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Numerical Verification of PolyDG Algebraic Solvers for the Pseudo-Stress Stokes Problem Antonietti, Paola F. Cancrini, Alessandra Ciaramella, Gabriele Numerical Analysis 65M60 This work focuses on the development of efficient solvers for the pseudo-stress formulation of the unsteady Stokes problem, discretised by means of a discontinuous Galerkin method on polytopal grids (PolyDG). The introduction of the pseudo-stress variable is motivated by the growing interest in non-Newtonian flow models and coupled interface problems, where the stress field plays a fundamental role in the physical description. The space-time discretisation of the problem is obtained by combining the PolyDG approach in space with the implicit Euler method for time integration. The resulting linear system, characterised by a symmetric, positive, definite matrix, exhibits deteriorating convergence with standard solvers as the time step decreases. To address this issue, we investigate two tailored strategies: deflated Conjugate Gradient, which mitigates the effect of the most problematic eigenmodes, and collective Block-Jacobi, which exploits the block structure of the system matrix. Numerical experiments show that both approaches yield iteration counts effectively independent of $Δt$, ensuring robust performance with respect to the time step. Future work will focus on extending this robustness to the spatial discretisation parameter $h$ by integrating multigrid strategies with the time-robust solvers developed in this study. |
| title | Numerical Verification of PolyDG Algebraic Solvers for the Pseudo-Stress Stokes Problem |
| topic | Numerical Analysis 65M60 |
| url | https://arxiv.org/abs/2512.02537 |