Stabilized Finite Element Approximation of the Mean Curvature Vector on Closed Surfaces

TL;DR

This paper introduces a stabilized finite element method for accurately approximating the mean curvature vector on closed surfaces, achieving first-order convergence and validated through theoretical analysis and numerical experiments.

Contribution

It develops a novel stabilized discrete Laplace-Beltrami operator with proven convergence properties for both standard and cut surfaces.

Findings

Achieves first-order convergence in L2 norm

Provides a priori error estimates

Numerical results confirm theoretical predictions

Abstract

We develop a stabilized discrete Laplace-Beltrami operator that is used to compute an approximate mean curvature vector which enjoys convergence of order one in L2. The stabilization is of gradient jump type and we consider both standard meshed surfaces and so called cut surfaces that are level sets of piecewise linear distance functions. We prove a priori error estimates and verify the theoretical results numerically.