Stabilized Lagrange Multipliers for Dirichlet Boundary Conditions in Divergence Preserving Unfitted Methods

TL;DR

This paper introduces a stabilized Lagrange multiplier method for unfitted finite element discretizations of the Darcy interface problem, achieving optimal convergence, well-posedness, and divergence-free velocity approximations.

Contribution

It develops a symmetric stabilized Lagrange multiplier approach with higher polynomial degree spaces to improve boundary condition enforcement in unfitted methods.

Findings

Achieves optimal convergence rates for velocity and pressure.

Ensures well-posed linear systems with favorable condition numbers.

Provides divergence-free velocity approximations.

Abstract

We extend the divergence preserving cut finite element method presented in [T. Frachon, P. Hansbo, E. Nilsson, S. Zahedi, SIAM J. Sci. Comput., 46 (2024)] for the Darcy interface problem to unfitted outer boundaries. We impose essential boundary conditions on unfitted meshes with a stabilized Lagrange multiplier method. The stabilization term for the Lagrange multiplier is important for stability but it may perturb the approximate solution at the boundary. We study different stabilization terms from cut finite element discretizations of surface partial differential equations and trace finite element methods. To reduce the perturbation we use a Lagrange multiplier space of higher polynomial degree compared to previous work on unfitted discretizations. We propose a symmetric method that results in 1) optimal rates of convergence for the approximate velocity and pressure; 2) well-posed linear systems where the condition number of the system matrix scales as for fitted finite element discretizations; 3) optimal approximation of the divergence with pointwise divergence-free approximations of solenoidal velocity fields. The three properties are proven to hold for the lowest order discretization and numerical experiments indicate that these properties continue to hold also when higher order elements are used.