An edge-based scheme on polyhedral meshes for vector advection-reaction equations

Pierre Cantin (1, 2) , Alexandre Ern (1, 3)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 EDF R&D - EDF R&D
3 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms

We devise and analyze an edge-based scheme on polyhedral meshes to approximate a vector advection-reaction problem. The well-posedness of the discrete problem is analyzed first under the classical positivity hypothesis of Friedrichs' systems that requires a lower bound on the lowest eigen-value of some tensor depending on the model parameters. We also prove stability when the lowest eigenvalue is null or even slightly negative if the mesh size is small enough. A priori error estimates are established for smooth solutions. Numerical results are presented on three-dimensional polyhedral meshes.

Mots clés: Friedrichs' assumptions; polyhedral meshes; Vector advection-reaction problems; quasi-optimal a priori error estimates

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

Pierre Cantin, Alexandre Ern. An edge-based scheme on polyhedral meshes for vector advection-reaction equations. ESAIM: Mathematical Modelling and Numerical Analysis, 2017, ⟨10.1051/m2an/2016075⟩. ⟨hal-01324545v3⟩