A Hybrid High-Order method for the Cahn-Hilliard problem in mixed form

Florent Chave (1) , Daniele Antonio Di Pietro (1) , Fabien Marche (1, 2) , Franck Pigeonneau (3)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 LEMON - Littoral, Environment: MOdels and Numerics
3 SVI - Surface du Verre et Interfaces

In this work, we develop a fully implicit Hybrid High-Order algorithm for the Cahn–Hilliard problem in mixed form. The space discretization hinges on local reconstruction operators from hybrid polynomial unknowns at elements and faces. The proposed method has several assets: (i) It supports fairly general meshes possibly containing polygonal elements and nonmatching interfaces; (ii) it allows arbitrary approximation orders; (iii) it has a moderate computational cost thanks to the possibility of locally eliminating element-based unknowns by static condensation. We perform a detailed stability and convergence study, proving optimal convergence rates in energy-like norms. Numerical validation is also provided using some of the most common tests in the literature.

Mots clés: polygonal meshes; phase separation; Hybrid High-Order; Cahn-Hilliard equation; mixed formulation; discrete functional analysis

Domaines: Analyse numérique [math.NA]

Florent Chave, Daniele Antonio Di Pietro, Fabien Marche, Franck Pigeonneau. A Hybrid High-Order method for the Cahn-Hilliard problem in mixed form. SIAM Journal on Numerical Analysis, 2016, 54 (3), pp.1873-1898. ⟨10.1137/15M1041055⟩. ⟨hal-01203733v2⟩