$\phi$-FEM, a finite element method on domains defined by level-sets: the Neumann boundary case

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.