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The mean curvature vector of a surface is obtained by letting the Laplace--Beltrami operator act on the embedding of the surface in ${\bf R}^3$. In this contribution we develop a stabilized finite element approximation of the mean curvature vector of certain piecewise linear surfaces which enjoys first order convergence in $L^2$. The stabilization involves the jump in the tangent gradient in the direction of the outer co-normals at each edge in the surface mesh. 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.

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