A High-Order, Pressure-Robust, and Decoupled Finite Difference Method for the Stokes Problem
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866909650192957440 |
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| author | Feng, Qiwei Han, Bin Neilan, Michael |
| author_facet | Feng, Qiwei Han, Bin Neilan, Michael |
| contents | In this paper, we consider the Stokes problem with Dirichlet boundary conditions and the constant kinematic viscosity $ν$ in an axis-aligned domain $Ω$. We decouple the velocity $\bm u$ and pressure $p$ by deriving a novel biharmonic equation in $Ω$ and third-order boundary conditions on $\partialΩ$. In contrast to the fourth-order streamfunction approach, our formulation does not require $Ω$ to be simply connected. For smooth velocity fields $\bm u$ in two dimensions, we explicitly construct a finite difference method (FDM) with sixth-order consistency to approximate $\bm u$ at all relevant grid points: interior points, boundary side points, and boundary corner points. The resulting scheme yields two linear systems $A_1u^{(1)}_h=b_1$ and $A_2u^{(2)}_h=b_2$, where $A_1,A_2$ are constant matrices, and $b_1,b_2$ are independent of the pressure $p$ and the kinematic viscosity $ν$. Thus, the proposed method is pressure- and viscosity-robust. To accommodate velocity fields with less regularity, we modify the FDM by removing singular terms in the right-hand side vectors. Once the discrete velocity is computed, we apply a sixth-order finite difference operator to approximate the pressure gradient locally, without solving any additional linear systems. In our numerical experiments, we test both smooth and non-smooth solutions $(\bm u,p)$ in a square domain, a triply connected domain, and an $L$-shaped domain in two dimensions. The results confirm sixth-order convergence of the velocity and pressure gradient in the $\ell_\infty$-norm for smooth solutions. For non-smooth velocity fields, our method achieves the expected lower-order convergence. Moreover, the observed velocity error $\|{\bm u}_h-\bm u\|_{\infty}$ is independent of the pressure $p$ and viscosity $ν$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13645 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A High-Order, Pressure-Robust, and Decoupled Finite Difference Method for the Stokes Problem Feng, Qiwei Han, Bin Neilan, Michael Numerical Analysis In this paper, we consider the Stokes problem with Dirichlet boundary conditions and the constant kinematic viscosity $ν$ in an axis-aligned domain $Ω$. We decouple the velocity $\bm u$ and pressure $p$ by deriving a novel biharmonic equation in $Ω$ and third-order boundary conditions on $\partialΩ$. In contrast to the fourth-order streamfunction approach, our formulation does not require $Ω$ to be simply connected. For smooth velocity fields $\bm u$ in two dimensions, we explicitly construct a finite difference method (FDM) with sixth-order consistency to approximate $\bm u$ at all relevant grid points: interior points, boundary side points, and boundary corner points. The resulting scheme yields two linear systems $A_1u^{(1)}_h=b_1$ and $A_2u^{(2)}_h=b_2$, where $A_1,A_2$ are constant matrices, and $b_1,b_2$ are independent of the pressure $p$ and the kinematic viscosity $ν$. Thus, the proposed method is pressure- and viscosity-robust. To accommodate velocity fields with less regularity, we modify the FDM by removing singular terms in the right-hand side vectors. Once the discrete velocity is computed, we apply a sixth-order finite difference operator to approximate the pressure gradient locally, without solving any additional linear systems. In our numerical experiments, we test both smooth and non-smooth solutions $(\bm u,p)$ in a square domain, a triply connected domain, and an $L$-shaped domain in two dimensions. The results confirm sixth-order convergence of the velocity and pressure gradient in the $\ell_\infty$-norm for smooth solutions. For non-smooth velocity fields, our method achieves the expected lower-order convergence. Moreover, the observed velocity error $\|{\bm u}_h-\bm u\|_{\infty}$ is independent of the pressure $p$ and viscosity $ν$. |
| title | A High-Order, Pressure-Robust, and Decoupled Finite Difference Method for the Stokes Problem |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2506.13645 |