Vertex-Based Compatible Discrete Operator Schemes on Polyhedral Meshes for Advection-Diffusion Equations

Pierre Cantin (1, 2) , Alexandre Ern (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 EDF R&D - EDF R&D

We devise and analyze vertex-based, Péclet-robust, lowest-order schemes for advection-diffusion equations that support polyhedral meshes. The schemes are formulated using Compatible Discrete Operators (CDO), namely primal and dual discrete differential operators, a discrete contraction operator for advection, and a discrete Hodge operator for diffusion. Moreover, discrete boundary operators are devised to weakly enforce Dirichlet boundary conditions. The analysis sheds new light on the theory of Friedrichs' operators at the purely algebraic level. Moreover, an extension of the stability analysis hinging on inf-sup conditions is presented to incorporate divergence-free velocity fields under some assumptions. Error bounds and convergence rates for smooth solutions are derived, and numerical results are presented on three-dimensional polyhedral meshes.

Mots clés: Mimetic discretization; Compatible schemes; Error Analysis; Polyhedral meshes; Advection schemes; Advection-Diffusion; Convection diffusion; Convection dominant regime; Advection equation

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Pierre Cantin, Alexandre Ern. Vertex-Based Compatible Discrete Operator Schemes on Polyhedral Meshes for Advection-Diffusion Equations. 2015. ⟨hal-01141290v1⟩