A divergence preserving cut finite element method for Darcy flow

TL;DR

This paper introduces a divergence-preserving cut finite element method for Darcy flow that maintains optimal convergence and stability, even with complex interface positioning, by employing new stabilization techniques.

Contribution

The authors develop a novel stabilization approach that preserves divergence-free properties in cut finite element methods for Darcy flow, improving accuracy and stability.

Findings

Achieves optimal convergence rates for velocity and pressure.

Ensures well-posed linear systems with controlled condition numbers.

Provides pointwise divergence-free velocity approximations.

Abstract

We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs $\textbf{RT}_k\times Q_k$, $k\geq 0$. Here $Q_k$ is the space of discontinuous polynomial functions of degree less or equal to $k$ and $\textbf{RT}$ is the Raviart-Thomas space. We show that the standard ghost penalty stabilization, often added in the weak forms of cut finite element methods for stability and control of the condition number of the linear system matrix, destroys the divergence-free property of the considered element pairs. Therefore, we propose new stabilization terms for the pressure and show that we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. We prove that the method with the new stabilization term has pointwise divergence-free approximations of solenoidal velocity fields. We derive a priori error estimates for the proposed unfitted finite element discretization based on $\textbf{RT}_k\times Q_k$, $k\geq 0$. In addition, by decomposing the mesh into macro-elements and applying ghost penalty terms only on interior edges of macro-elements, stabilization is applied very restrictively and only where needed. Numerical experiments with element pairs $\textbf{RT}_0\times Q_0$, $\textbf{RT}_1\times Q_1$, and $\textbf{BDM}_1\times Q_0$ (where $\textbf{BDM}$ is the Brezzi-Douglas-Marini space) indicate that we have 1) optimal rates of convergence of the approximate velocity and pressure; 2) well-posed linear systems where the condition number of the system matrix scales as it does for fitted finite element discretizations; 3) optimal rates of convergence of the approximate divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative to the computational mesh.