Hybridization of Mixed High-Order Methods on General Meshes and Application to the Stokes Equations

Joubine Aghili (1) , Sébastien Boyaval (2, 3) , Daniele Di Pietro (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 LHSV - Laboratoire d'Hydraulique Saint-Venant / Saint-Venant Laboratory for Hydraulics
3 MATHERIALS - MATHematics for MatERIALS

This paper presents two novel contributions on the recently introduced Mixed High-Order (MHO) methods [`Arbitrary order mixed methods for heterogeneous anisotropic diffusion on general meshes', preprint (2013)]. We first address the hybridization of the MHO method for a scalar diffusion problem and obtain the corresponding primal formulation. Based on the hybridized MHO method, we then design a novel, arbitrary order method for the Stokes problem on general meshes. A full convergence analysis is carried out showing that, when independent polynomials of degree k are used as unknowns (at elements for the pressure and at faces for each velocity component), the energy-norm of the velocity and the L2-norm of the pressure converge with order (k + 1), while the L2-norm of the velocity (super-)converges with order (k + 2). The latter property is not shared by other methods based on a similar choice of unknowns. The theoretical results are numerically validated in two space dimensions on both standard and polygonal meshes.

Mots clés: General Meshes; Mixed High-Order Methods; Stokes; Hybridization

Domaines: Analyse numérique [math.NA]

Joubine Aghili, Sébastien Boyaval, Daniele Di Pietro. Hybridization of Mixed High-Order Methods on General Meshes and Application to the Stokes Equations. Computational Methods in Applied Mathematics, 2015, ⟨10.1515/cmam-2015-0004⟩. ⟨hal-01009723v2⟩