A discontinuous-skeletal method for advection-diffusion-reaction on general meshes
For computational scientists solving advection-diffusion-reaction problems, this method offers a flexible, accurate, and efficient approach on complex meshes.
The paper introduces a discontinuous-skeletal method for advection-diffusion-reaction equations using face-based polynomial degrees of freedom, supporting general meshes and arbitrary polynomial orders. The method achieves robust error analysis across all Péclet numbers, including degenerate diffusion, with moderate computational costs.
We design and analyze an approximation method for advection-diffusion-reaction equations 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 polytopal 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.