phi-FEM for the heat equation: optimal convergence on unfitted meshes in space

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

1 MIMESIS - Computational Anatomy and Simulation for Medicine
2 I2M - Institut de Mathématiques de Marseille
3 IMAG - Institut Montpelliérain Alexander Grothendieck
4 IMAG - Institut Montpelliérain Alexander Grothendieck
5 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

Thanks to a finite element method, we solve numerically parabolic partial differential equations on complex domains by avoiding the mesh generation, using a regular background mesh, not fitting the domain and its real boundary exactly. Our technique follows the phi-FEM paradigm, which supposes that the domain is given by a level-set function. In this paper, we prove a priori error estimates in L 2 (H 1) and L ∞ (L 2) norms for an implicit Euler discretization in time. We give numerical illustrations to highlight the performances of phi-FEM, which combines optimal convergence accuracy, easy implementation process and fastness.

Mots clés: finite element method; heat equation; fictitious domain; level-set; Dirichlet conditions; Numerical analysis

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

Michel Duprez, Vanessa Lleras, Alexei Lozinski, Killian Vuillemot. phi-FEM for the heat equation: optimal convergence on unfitted meshes in space. Comptes Rendus. Mathématique, 2023, 361 (G11), pp.1699-1710. ⟨10.5802/crmath.497⟩. ⟨hal-03685445v2⟩