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

Michel Duprez (1, 2) , Vanessa Lleras (3) , Alexei Lozinski (4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 MIMESIS - Computational Anatomy and Simulation for Medicine
3 IMAG - Institut Montpelliérain Alexander Grothendieck
4 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

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.

Mots clés: Neumann conditions; Finite element method; fictitious domain; level-set

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Michel Duprez, Vanessa Lleras, Alexei Lozinski. φ-FEM, a finite element method on domains defined by level-sets: the Neumann boundary case. 2020. ⟨hal-02521042v1⟩