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

TL;DR

This work develops a divergence-preserving CutFEM for the Darcy problem with Dirichlet boundary conditions on unfitted boundaries by using a stabilized Lagrange multiplier to weakly enforce $u\cdot n = u_B$ on $\Sigma$. A higher-degree multiplier space $Q_{h,k+1}^{\Sigma}$ is employed to mitigate boundary-perturbation effects, yielding a symmetric, well-conditioned system with optimal convergence for the lowest-order discretization and preserving pointwise divergence. The authors establish divergence-preserving properties, stability via an inf-sup framework, interpolation and consistency estimates, and a priori error bounds, and they prove a condition number scaling of $\kappa(A) \lesssim h^{-2}$. Numerical experiments across three scenarios (including unfitted, fitted, and interface problems) confirm the theoretical results and demonstrate robustness to higher-order elements and alternative stabilization choices, indicating practical utility for unfitted Darcy-type simulations.

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.

Figures (11)

• Figure 2.1: Illustration of the domains $\Omega\subset\Omega_{\mathcal{T}}\subset\Omega_0$, the active mesh $\mathcal{T}_h$ (triangles in grey), and faces in $\mathcal{F}_{\Sigma}$ (faces marked in yellow). The gray colored domain is $\Omega_{\mathcal{T}}$.
• Figure 5.1: Example 1: Magnitude of the approximated velocity field with the element triple $\mathbf{RT}_1\times Q_1\times Q_2^{\Sigma}$.
• Figure 5.2: Example 1: The divergence error versus mesh size $h$, using element triples $\mathbf{RT}_0\times Q_0 \times Q^{\Sigma}_1$ and $\mathbf{RT}_1\times Q_1\times Q^{\Sigma}_2$. The different stabilization terms for the Lagrange multiplier are compared. Left: The $L^2$-error of the divergence. Right: The pointwise error of the divergence.
• Figure 5.3: Example 1: Convergence and condition numbers using element triples $\mathbf{RT}_0\times Q_0 \times Q^{\Sigma}_1$ and $\mathbf{RT}_1\times Q_1\times Q^{\Sigma}_2$ against mesh size $h$. The different stabilization terms for the Lagrange multiplier are compared. Left: The $L^2$-error of the pressure. Middle: The $L^2$-error of the velocity field. Right: The $1$-norm condition number.
• Figure 5.4: Example 1.1: Convergence and condition numbers versus mesh size $h$ using element triple $\mathbf{RT}_1\times Q_1\times Q^{\Sigma}_1$ and $\mathbf{RT}_1\times Q_1\times Q^{\Sigma}_2$. The alternative stabilization terms $\hat{s}_c$ and $\tilde{s}_c$ are compared. Left: The $L^2$-error of the pressure. Middle: The $L^2$-error of the velocity field. Right: The $1$-norm condition number.

Theorems & Definitions (34)

• Remark 2.1: Stabilization
• Remark 2.2: The approximation space for the discrete Lagrange multiplier
• Remark 2.3: The linear system
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4