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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
* Corresponding author
3 MATHERIALS - MATHematics for MatERIALS
CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique, Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech
Abstract : 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.
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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, De Gruyter, 2015, ⟨10.1515/cmam-2015-0004⟩. ⟨hal-01009723v2⟩

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