We extend a fictitious domain-type finite element method, called φ-FEM and introduced in [7], 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 φ-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：水平集定义域的有限元方法

我们将在 [7] 中引入的称为 φ-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 φ-FEM 的最佳收敛性以及离散问题独立于网格切割的良好条件这一事实。数值实验证实了理论结果。