An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators

Daniele Antonio Di Pietro (1) , Alexandre Ern (2) , Simon Lemaire (2)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

We develop an arbitrary-order primal method for diffusion problems on general polyhedral meshes. The degrees of freedom are scalar-valued polynomials of the same order at mesh elements and faces. The cornerstone of the method is a local (element-wise) discrete gradient reconstruction operator. The design of the method additionally hinges on a least-squares penalty term on faces weakly enforcing the matching between local element- and face-based degrees of freedom. The scheme is proved to optimally converge in the energy norm and in the $L^2$-norm of the potential for smooth solutions. In the lowest-order case, equivalence with the Hybrid Finite Volume method is shown. The theoretical results are confirmed by numerical experiments up to order 4 on several polygonal meshes.

Mots clés: Diffusion; General meshes; Arbitrary-order methods; Gradient reconstruction

Domaines: Analyse numérique [math.NA]

Daniele Antonio Di Pietro, Alexandre Ern, Simon Lemaire. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Computational Methods in Applied Mathematics, 2014, 14 (4), pp.461-472. ⟨10.1515/cmam-2014-0018⟩. ⟨hal-00978198v2⟩