Stabilization of High Order Cut Finite Element Methods on Surfaces

TL;DR

This paper introduces a new stabilization technique for high-order cut finite element methods on surfaces, ensuring well-conditioned systems and optimal error estimates, validated through theoretical analysis and numerical experiments.

Contribution

A novel stabilization term for cut finite element methods on surfaces that guarantees optimal condition number scaling and error estimates for both linear and higher-order elements.

Findings

Condition number of stiffness matrix is O(h^{-2})

Stabilization works for linear and higher-order elements

Numerical results confirm theoretical predictions

Abstract

We develop and analyze a stabilization term for cut finite element approximations of an elliptic second order partial differential equation on a surface embedded in $\mathbb{R}^d$. The new stabilization term combines properly scaled normal derivatives at the surface together with control of the jump in the normal derivatives across faces and provides control of the variation of the finite element solution on the active three dimensional elements that intersect the surface. We show that the condition number of the stiffness matrix is $O(h^{-2})$, where $h$ is the mesh parameter. The stabilization term works for linear as well as for higher-order elements and the derivation of its stabilizing properties is quite straightforward, which we illustrate by discussing the extension of the analysis to general $n$-dimensional smooth manifolds embedded in $\mathbb{R}^d$, with codimension $d-n$. We also formulate properties of a general stabilization term that are sufficient to prove optimal scaling of the condition number and optimal error estimates in energy- and $L^2$-norm. We finally present numerical studies confirming our theoretical results.