An unfitted divergence-free higher order finite element method for the Stokes problem

The paper develops and analyzes a higher-order unfitted finite element method for the incompressible Stokes equations, which yields a strongly divergence-free velocity field up to the physical boundary. The method combines an isoparametric Scott–Vogelius velocity-pressure pair on a cut background mesh with a stabilized Nitsche/Lagrange multiplier formulation for imposing Dirichlet boundary conditions. We construct finite element spaces that admit robust numerical implementation using standard elementwise polynomial mappings and produce exactly divergence-free discrete velocities. The key components of the analysis are a new inf-sup stability result for the isoparametric Scott–Vogelius pair on unfitted meshes and a combined inf-sup stability result for the bilinear forms associated with the pressure and the Lagrange multiplier. The finite element formulation employs a higher-order Lagrange multiplier space, which ensures stability and mitigates the loss of pressure robustness typically associated with the weak enforcement of boundary conditions for the normal velocity component. The paper provides a complete stability and convergence theory in two dimensions, accounting for the geometric errors introduced by the isoparametric approximation. The analysis demonstrates optimal-order velocity convergence in both the $H^1$ and $L^2$ norms and establishes optimal $H^1$-convergence and nearly optimal $L^2$-convergence of a post-processed pressure. Numerical experiments illustrate and confirm the theoretical findings.

💡 Research Summary

This paper presents a sophisticated advancement in computational fluid dynamics, specifically addressing the challenges of solving the incompressible Stokes equations using an unfitted finite element method (FEM). The primary motivation behind this research is the need for a robust numerical framework that can handle complex, evolving geometries without the computationally expensive process of remeshing, while simultaneously maintaining the fundamental physical principle of mass conservation—the divergence-free constraint of the velocity field.

The core innovation of this work lies in the development of a higher-order unfitted FEM that ensures a strongly divergence-free velocity field up to the physical boundary. To achieve this, the authors employ an isoparametric Scott-Vogelius velocity-pressure element pair on a cut background mesh. While the Scott-Vogelius element is renowned for its divergence-free properties on fitted meshes, extending this property to unfitted meshes is non-trivial due to the geometric irregularities introduced by the boundary cutting through elements. The authors overcome this by utilizing isoparametric polynomial mappings, which allow for a high-order approximation of the boundary geometry, thereby mitigating geometric errors.

A significant technical hurdle in unfitted methods is the potential loss of pressure robustness and numerical instability when imposing Dirichlet boundary conditions weakly. The authors address this by implementing a stabilized formulation that combines Nitsche’s method with a Lagrange multiplier approach. Crucially, they utilize a higher-order Lagrange multiplier space. This strategic choice is vital to ensure the stability of the pressure field and to prevent the degradation of accuracy that typically occurs in standard unfitted formulations when the normal velocity component is weakly enforced.

The mathematical rigor of the paper is demonstrated through a comprehensive stability analysis. The authors establish a new inf-sup (LBB) stability result for the isoparametric Scott-Vogelius pair on unfitted meshes. Furthermore, they provide a combined inf-sup stability proof for the bilinear forms involving the pressure and the Lagrange multiplier. This theoretical foundation guarantees that the discrete system is well-posed and stable.

In terms of convergence, the paper provides a complete theory in two dimensions. The analysis proves that the velocity field achieves optimal-order convergence in both the $H^1$ and $L^2$ norms. For the pressure, the authors demonstrate optimal $H^1$-convergence and nearly optimal $L^2$-convergence through a post-processing technique. Numerical experiments presented in the paper validate these theoretical findings, showing that the proposed method maintains high accuracy and stability even in the presence of complex boundary intersections. This research provides a powerful and mathematically sound tool for simulating incompressible flows in complex, time-varying domains.