φ-FEM: an optimally convergent and easily implementable immersed boundary method for particulate flows and Stokes equations

Michel Duprez (1, 2, 3) , Vanessa Lleras (4) , Alexei Lozinski (5)

1 MIMESIS - Computational Anatomy and Simulation for Medicine
2 ICube - Laboratoire des sciences de l'ingénieur, de l'informatique et de l'imagerie
3 Inria Nancy - Grand Est
4 IMAG - Institut Montpelliérain Alexander Grothendieck
5 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

We present an immersed boundary method to simulate the creeping motion of a rigid particle in a fluid described by the Stokes equations discretized thanks to a finite element strategy on unfitted meshes, called φ-FEM, that uses the description of the solid with a level-set function. One of the advantages of our method is the use of standard finite element spaces and classical integration tools, while maintaining the optimal convergence (theoretically in the H¹ norm for the velocity and L² for pressure; numerically also in the L² norm for the velocity).

Mots clés: Finite element method; numerical analysis; fictitious domain; immersed boundary method; level-set

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

Michel Duprez, Vanessa Lleras, Alexei Lozinski. φ-FEM: an optimally convergent and easily implementable immersed boundary method for particulate flows and Stokes equations. ESAIM: Mathematical Modelling and Numerical Analysis, 2023, 57, pp.1111-1142. ⟨10.1051/m2an/2023010⟩. ⟨hal-03588715v4⟩