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We propose and analyze a new stabilized cut finite element method for theLaplace-Beltrami operator on a closed surface. The new stabilization termprovides control of the full $\mathbb{R}^3$ gradient on the active meshconsisting of the elements that intersect the surface. Compared to facestabilization, based on controlling the jumps in the normal gradient acrossfaces between elements in the active mesh, the full gradient stabilization iseasier to implement and does not significantly increase the number of nonzeroelements in the mass and stiffness matrices. The full gradient stabilizationterm may be combined with a variational formulation of the Laplace-Beltramioperator based on tangential or full gradients and we present a simple andunified analysis that covers both cases. The full gradient stabilization termgives rise to a consistency error which, however, is of optimal order forpiecewise linear elements, and we obtain optimal order a priori error estimatesin the energy and $L^2$ norms as well as an optimal bound of the conditionnumber. Finally, we present detailed numerical examples where we in particularstudy the sensitivity of the condition number and error on the stabilizationparameter.

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