Cut Finite Element Methods for Partial Differential Equations on Embedded Manifolds of Arbitrary Codimensions

TL;DR

This paper introduces a stabilized cut finite element method for solving PDEs like the Laplace-Beltrami operator on manifolds embedded in higher-dimensional spaces, accommodating arbitrary codimensions and non-matching meshes.

Contribution

It develops a theoretical framework with stability and error analysis for cut finite element methods on embedded manifolds of any codimension, including novel stabilization techniques.

Findings

Proven bounds on condition numbers of the stiffness matrix

Achieved optimal a priori error estimates

Numerical validation on curves and surfaces in 3D

Abstract

We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in $\mathbb{R}^d$ of arbitrary codimension. The method is based on using continuous piecewise polynomials on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in $\mathbb{R}^3$.