A cut finite element method for the Darcy problem

TL;DR

This paper introduces a cut finite element method for the Darcy problem that employs Nitsche's formulation and ghost penalty stabilization, providing theoretical analysis and numerical validation for velocity and pressure accuracy.

Contribution

It extends the cut finite element method to the Darcy problem with Neumann boundary conditions, incorporating ghost penalties for stabilization and providing rigorous error analysis.

Findings

Method is well-posed with proven stability.

Error estimates for velocity and pressure are derived.

Numerical results confirm theoretical predictions.

Abstract

We present and analyze a cut finite element method for the weak imposition of the Neumann boundary conditions of the Darcy problem. The Raviart-Thomas mixed element on both triangular and quadrilateral meshes is considered. Our method is based on the Nitsche formulation studied in [10.1515/jnma-2021-0042] and can be considered as a first attempt at extension in the unfitted case. The key feature is to add two ghost penalty operators to stabilize both the velocity and pressure fields. We rigorously prove our stabilized formulation to be well-posed and derive a priori error estimates for the velocity and pressure fields. We show that an upper bound for the condition number of the stiffness matrix holds as well. Numerical examples corroborating the theory are included.