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

TL;DR

This paper introduces a novel discontinuous-skeletal approximation method for advection-diffusion-reaction equations on complex meshes, supporting arbitrary polynomial orders and robustly handling various Péclet numbers, including degenerate cases.

Contribution

The paper develops a new discontinuous-skeletal method with local reconstruction operators for general meshes, extending applicability to degenerate diffusion scenarios and arbitrary polynomial degrees.

Findings

Supports general polytopal and nonmatching meshes.

Handles full Péclet number range, including degenerate cases.

Maintains moderate computational costs with compact stencils.

Abstract

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 P\'eclet 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.