Abstract
Abstract We develop and analyse a stabilization term for cut finite element approximations of an elliptic second-order partial differential equation on a surface embedded in ${\mathbb{R}}^d$. The new stabilization term combines properly scaled normal derivatives at the surface together with control of the jump in the normal derivatives across faces, and provides control of the variation of the finite element solution on the active three-dimensional elements that intersect the surface. We show that the condition number of the stiffness matrix is $O(h^{-2})$, where $h$ is the mesh parameter. The stabilization term works for linear as well as for higher-order elements and the derivation of its stabilizing properties is quite straightforward, which we illustrate by discussing the extension of the analysis to general $n$-dimensional smooth manifolds embedded in ${\mathbb{R}}^d$, with codimension $d-n$. We also state the properties of a general stabilization term that are sufficient to prove optimal scaling of the condition number and optimal error estimates in energy- and $L^2$-norm. We finally present numerical studies confirming our theoretical results.
Highlights
Cut finite element method for surface partial differential equations
We formulate properties of a general stabilization term that are sufficient to prove that the resulting linear system of equations have an optimal scaling of the condition number with the mesh parameter, and that the convergence of the proposed method is of optimal order
We compare the proposed stabilization with the stabilization term (1.10), and in the computation of the latter stabilization term normal vectors at quadrature points inside elements cut by the interface are needed
Summary
Cut finite element method for surface partial differential equations. CutFEM Where γ ∈ [0, 1] is a parameter, [DnjF v] is the jump in the normal derivative of order j across the face F and cF,j > 0, cΓ ,j > 0, are stabilization constants This stabilization term with γ = 1 has been used with a high order space-time CutFEM for PDEs on evolving surfaces by Zahedi (2018).
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