Full Gradient Stabilized Cut Finite Element Methods for Surface Partial Differential Equations

TL;DR

This paper introduces a new stabilized cut finite element method for surface PDEs that controls the full gradient, offering easier implementation and optimal error estimates for Laplace-Beltrami problems.

Contribution

The paper develops a full gradient stabilization technique for cut finite element methods on surfaces, providing a unified analysis and optimal error estimates.

Findings

Full gradient stabilization is easier to implement than face stabilization.

The method achieves optimal order error estimates in energy and L2 norms.

Numerical experiments demonstrate the method's robustness and sensitivity to parameters.

Abstract

We propose and analyze a new stabilized cut finite element method for the Laplace-Beltrami operator on a closed surface. The new stabilization term provides control of the full $\mathbb{R}^3$ gradient on the active mesh consisting of the elements that intersect the surface. Compared to face stabilization, based on controlling the jumps in the normal gradient across faces between elements in the active mesh, the full gradient stabilization is easier to implement and does not significantly increase the number of nonzero elements in the mass and stiffness matrices. The full gradient stabilization term may be combined with a variational formulation of the Laplace-Beltrami operator based on tangential or full gradients and we present a simple and unified analysis that covers both cases. The full gradient stabilization term gives rise to a consistency error which, however, is of optimal order for piecewise linear elements, and we obtain optimal order a priori error estimates in the energy and $L^2$ norms as well as an optimal bound of the condition number. Finally, we present detailed numerical examples where we in particular study the sensitivity of the condition number and error on the stabilization parameter.