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

TL;DR

This paper introduces a higher-order unfitted finite element method for the Stokes problem that produces exactly divergence-free velocities and demonstrates optimal convergence properties through rigorous analysis and numerical validation.

Contribution

It develops a novel higher-order unfitted finite element approach with stability and convergence analysis, ensuring divergence-free velocities and accurate pressure approximation.

Findings

Achieves optimal velocity convergence in $H^1$ and $L^2$ norms.

Ensures exactly divergence-free discrete velocities.

Numerical experiments confirm theoretical convergence rates.

Abstract

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.