Cut finite element method for divergence free approximation of incompressible flow: a Lagrange multiplier approach

TL;DR

This paper introduces a cut finite element method for incompressible flow that ensures divergence-free velocity approximation using a Lagrange multiplier, with optimal error estimates independent of domain-mesh intersection complexities.

Contribution

It develops a novel cut finite element approach with divergence-free velocities for Stokes problems, employing Nitsche's or stabilized Lagrange multipliers for boundary conditions.

Findings

Divergence of velocities is pointwise zero across the mesh.

Optimal error estimates are derived for velocity and pressure.

Error constants are independent of domain-mesh intersection and pressure regularity.

Abstract

In this note we design a cut finite element method for a low order divergence free element applied to a boundary value problem subject to Stokes' equations. For the imposition of Dirichlet boundary conditions we consider either Nitsche's method or a stabilized Lagrange multiplier method. In both cases the normal component of the velocity is constrained using a multiplier, different from the standard pressure approximation. The divergence of the approximate velocities is pointwise zero over the whole mesh domain, and we derive optimal error estimates for the velocity and pressures, where the error constant is independent of how the physical domain intersects the computational mesh, and of the regularity of the pressure multiplier imposing the divergence free condition.