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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.22077 |
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| _version_ | 1866915576483414016 |
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| author | Li, Kangan Mehmani, Yashar |
| author_facet | Li, Kangan Mehmani, Yashar |
| contents | We aim to solve the incompressible Navier-Stokes equations within the complex microstructure of a porous material. Discretizing the equations on a fine grid using a staggered (e.g., marker-and-cell, mixed FEM) scheme results in a nonlinear residual. Adopting the Newton method, a linear system must be solved at each iteration, which is large, ill-conditioned, and has a saddle-point structure. This demands an iterative (e.g., Krylov) solver, that requires preconditioning to ensure rapid convergence. We propose two monolithic \textit{algebraic} preconditioners, $a\mathrm{PLMM_{NS}}$ and $a\mathrm{PNM_{NS}}$, that are generalizations of previously proposed forms by the authors for the Stokes equations ($a\mathrm{PLMM_{S}}$ and $a\mathrm{PNM_{S}}$). The former is based on the pore-level multiscale method (PLMM) and the latter on the pore network model (PNM), both successful approximate solvers. We also formulate faster-converging geometric preconditioners $g\mathrm{PLMM}$ and $g\mathrm{PNM}$, which impose $\partial_n\boldsymbol{u}\!=\!0$ (zero normal-gradient of velocity) exactly at subdomain interfaces. Finally, we propose an accurate coarse-scale solver for the steady-state Navier-Stokes equations based on $g\mathrm{PLMM}$, capable of computing approximate solutions orders of magnitude faster. We benchmark our preconditioners against state-of-the-art block preconditioners and show $g\mathrm{PLMM}$ is the best-performing one, followed closely by $a\mathrm{PLMM_{S}}$ for steady-state flow and $a\mathrm{PLMM_{NS}}$ for transient flow. All preconditioners can be built and applied on parallel machines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22077 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Preconditioning and Reduced-Order Modeling of Navier-Stokes Equations in Complex Porous Microstructures Li, Kangan Mehmani, Yashar Numerical Analysis Computational Physics We aim to solve the incompressible Navier-Stokes equations within the complex microstructure of a porous material. Discretizing the equations on a fine grid using a staggered (e.g., marker-and-cell, mixed FEM) scheme results in a nonlinear residual. Adopting the Newton method, a linear system must be solved at each iteration, which is large, ill-conditioned, and has a saddle-point structure. This demands an iterative (e.g., Krylov) solver, that requires preconditioning to ensure rapid convergence. We propose two monolithic \textit{algebraic} preconditioners, $a\mathrm{PLMM_{NS}}$ and $a\mathrm{PNM_{NS}}$, that are generalizations of previously proposed forms by the authors for the Stokes equations ($a\mathrm{PLMM_{S}}$ and $a\mathrm{PNM_{S}}$). The former is based on the pore-level multiscale method (PLMM) and the latter on the pore network model (PNM), both successful approximate solvers. We also formulate faster-converging geometric preconditioners $g\mathrm{PLMM}$ and $g\mathrm{PNM}$, which impose $\partial_n\boldsymbol{u}\!=\!0$ (zero normal-gradient of velocity) exactly at subdomain interfaces. Finally, we propose an accurate coarse-scale solver for the steady-state Navier-Stokes equations based on $g\mathrm{PLMM}$, capable of computing approximate solutions orders of magnitude faster. We benchmark our preconditioners against state-of-the-art block preconditioners and show $g\mathrm{PLMM}$ is the best-performing one, followed closely by $a\mathrm{PLMM_{S}}$ for steady-state flow and $a\mathrm{PLMM_{NS}}$ for transient flow. All preconditioners can be built and applied on parallel machines. |
| title | Preconditioning and Reduced-Order Modeling of Navier-Stokes Equations in Complex Porous Microstructures |
| topic | Numerical Analysis Computational Physics |
| url | https://arxiv.org/abs/2510.22077 |