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We develop and analyze a stabilization term for cut finite elementapproximations of an elliptic second order partial differential equation on asurface embedded in $\mathbb{R}^d$. The new stabilization term combinesproperly scaled normal derivatives at the surface together with control of thejump in the normal derivatives across faces and provides control of thevariation of the finite element solution on the active three dimensionalelements that intersect the surface. We show that the condition number of thestiffness matrix is $O(h^{-2})$, where $h$ is the mesh parameter. Thestabilization term works for linear as well as for higher-order elements andthe derivation of its stabilizing properties is quite straightforward, which weillustrate 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 aresufficient to prove optimal scaling of the condition number and optimal errorestimates in energy- and $L^2$-norm. We finally present numerical studiesconfirming our theoretical results.

Stabilization of High Order Cut Finite Element Methods on Surfaces