Ghost stabilisation for cut finite element exterior calculus

Abstract

We introduce the cut finite element method in the language of finite element exterior calculus, by formulating a stabilisation -- for any form degree -- that makes the method robust with respect to the position of the interface relative to the mesh. We prove that the $L^2$-norm on the physical domain augmented with this stabilisation is uniformly equivalent to the $L^2$-norm on the ``active'' mesh that contains all the degrees of freedom of the finite element space (including those external to the physical domain). We show how this CutFEEC method can be applied to discretize the Hodge Laplace equations on an unfitted mesh, in any dimension and any topology, and prove stability and optimal convergence. Two numerical examples are provided, with convergence and condition number scaling independent of the position of the boundary with respect to the background mesh. We first solve the Hodge Laplace equation on a conforming finite element space of $H^{\text{curl}}$ posed on a filled torus. The second example extends beyond the Hodge Laplace problem and illustrates the importance of stabilisation for pressure robustness in unfitted schemes, using an $H^{\text{curl}}$-formulation for the Stokes equations.