We extend a fictitious domain-type finite element method, called $\phi$-FEM and introduced in arXiv:1903.03703 [math.NA], to the case of Neumann boundary conditions. The method is based on a multiplication by the level-set function and does not require a boundary fitted mesh. Unlike other recent fictitious domain-type methods (XFEM, CutFEM), our approach does not need any non-standard numerical integration on cut mesh elements or on the actual boundary. We prove the optimal convergence of $\phi$-FEM and the fact that the discrete problem is well conditioned inependently of the mesh cuts. The numerical experiments confirm the theoretical results.

中文翻译：

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$\phi$-FEM，由水平集定义的域上的有限元方法：Neumann 边界情况

我们将在 arXiv:1903.03703 [math.NA] 中引入的称为 $\phi$-FEM 的虚拟域类型有限元方法扩展到 Neumann 边界条件的情况。该方法基于与水平集函数的乘法，不需要边界拟合网格。与其他最近虚构的域类型方法（XFEM、CutFEM）不同，我们的方法不需要对切割网格元素或实际边界进行任何非标准的数值积分。我们证明了 $\phi$-FEM 的最佳收敛性以及离散问题的良好条件与网格切割无关。数值实验证实了理论结果。