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

Abstract : We design and analyze an approximation method for advection-diffusion-reaction equa-tions where the (generalized) degrees of freedom are polynomials of order $k\ge0$ 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.
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Daniele Antonio Di Pietro, Jerome Droniou, Alexandre Ern. A discontinuous-skeletal method for advection-diffusion-reaction on general meshes. SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2015, 53 (5), pp.2135-2157. ⟨10.1137/140993971⟩. ⟨hal-01079342v3⟩

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