A divergence-free parametric finite element method for 3D Stokes equations on curved domains

TL;DR

This work introduces a high-order divergence-free parametric mixed DG method for the 3D Stokes equations on curved domains. By discretizing velocity with parametric BDM elements and pressure with compatible volume elements, and employing an interior-penalty DG formulation, the authors establish stability via an inf-sup condition and achieve optimal convergence, with the notable feature that the discrete velocity is exactly divergence-free. The analysis leverages a smooth divergence-free extension of the exact solution and a domain transformation to the computational mesh, enabling rigorous error estimates. Numerical experiments on curved versus straight meshes and a 3D cavity with a spherical void validate the theory and demonstrate practical accuracy and robustness on curved geometries.

Abstract

The Stokes equations play an important role in the incompressible flow simulation. In this paper, a novel divergence-free parametric mixed finite element method is proposed for solving three-dimensional Stokes equations on domains with piecewise smooth boundaries. The flow velocity and pressure are discretized with high-order parametric Brezzi-Douglas-Marini elements and volume elements, respectively, on curved tetrahedral meshes. Utilizing the interior-penalty discontinuous Galerkin (IPDG) technique, we prove the inf-sup condition for the mixed finite element pair, and high-order optimal error estimates in the energy norm, with the help of the extension and transformation of the true solution to computational domain. Moreover, the discrete velocity is exactly divergence-free, meaning that div uh = 0 holds in the curved computational domain. Numerical experiments are conducted to support the theoretical analyses.

Figures (6)

• Figure 1: Left: A curved tetrahedron. Right: A sphere domain divided by curved tetrahedra.
• Figure 2: The decreasing trend of errors for ${\boldsymbol{u}}_h$ on both the straight meshes and curved meshes.
• Figure 3: The decreasing trend of errors for $p_h$ on both the straight meshes and curved meshes.
• Figure 4: The 3D distribution of velocity ${\boldsymbol{u}}_h$ with 27,889 tetrahedra, where the flow directions are plotted after two clips of $y = 0.5$ and $z = 0.85$.
• Figure 5: Streamlines of ${\boldsymbol{u}}_h$ from the line source $(0,0.5,0.8)\leftrightarrow(1,0.5,0.8)$

Theorems & Definitions (16)

• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Theorem 2.8
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3