A discontinuous-skeletal method for advection-diffusion-reaction on general meshes

Daniele Antonio Di Pietro (1) , Jerome Droniou (2) , Alexandre Ern (3)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 School of Mathematical Sciences [Clayton]
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

We design and analyze an approximation method for advection-diffusion-reaction equa-tions where the (generalized) degrees of freedom are polynomials of order k>=0 at mesh faces. The method hinges on local discrete reconstruction operators for the diffusive and advective derivatives and a weak enforcement of boundary conditions. Fairly general meshes with poly-topal and nonmatching cells are supported. Arbitrary polynomial orders can be considered, including the case k=0 which is closely related to Mimetic Finite Difference/Mixed-Hybrid Finite Volume methods. The error analysis covers the full range of Péclet numbers, including the delicate case of local degeneracy where diffusion vanishes on a strict subset of the domain. Computational costs remain moderate since the use of face unknowns leads to a compact stencil with reduced communications. Numerical results are presented.

Mots clés: discontinuous skeletal methods; advection-diffusion-reaction; locally degenerate diffusion

Domaines: Analyse numérique [math.NA]

Daniele Antonio Di Pietro, Jerome Droniou, Alexandre Ern. A discontinuous-skeletal method for advection-diffusion-reaction on general meshes. 2014. ⟨hal-01079342v1⟩