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

Cantin, Pierre; Ern, Alexandre

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1561-1581

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 eigenvalue 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 solutions in ${𝐖}^{1,q}\left(\Omega \right)$ with q ∈ ((3/2),2]. Numerical results are presented on three-dimensional polyhedral meshes.

Classification : 65N12, 65N15, 76R99, 76D07, 76W05
Keywords: Vector advection-reaction problems, polyhedral meshes, Friedrichs’ assumptions, quasi-optimala priori error estimates

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