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

TL;DR

This paper introduces a high-order divergence-free parametric finite element method for 3D Stokes equations on curved domains, ensuring exact incompressibility and providing optimal error estimates with numerical validation.

Contribution

A novel divergence-free parametric mixed finite element method for 3D Stokes equations on curved domains, with proven stability and optimal error bounds.

Findings

Discrete velocity is exactly divergence-free.

Method achieves high-order optimal error estimates.

Numerical experiments confirm theoretical results.

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.