An Incidence-Based Edge-Averaged Method for Convection-Diffusion Problems on Polytopal Meshes
Abstract
We develop an incidence-based edge-averaged method for scalar convection-diffusion problems on convex polytopal meshes in arbitrary dimensions. The proposed method extends the classical edge-averaged finite-element scheme through a graph-theoretic formulation based on vertex-edge incidence matrices. We first establish this formulation on simplices in arbitrary dimensions and then extend it to convex polytopes by introducing complementary edges and generalized barycentric coordinates. A key ingredient is a matrix identity that relates the generalized Whitney edge mass matrix to a diagonal representation of the Poisson stiffness matrix, enabling an exponentially-fitted discretization expressed entirely in terms of nodal values and edge weights. We prove that the resulting stiffness matrix is a nonsingular M-matrix under suitable positivity and connectivity conditions, ensuring monotonicity and the discrete maximum principle. Numerical experiments on two- and three-dimensional simplicial and nonsimplicial meshes demonstrate first-order convergence in the H1 semi-norm and stable, nonoscillatory solutions in convection-dominated regimes.
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BibTeX
@misc{adler2026incidencebased,
title = {{An Incidence-Based Edge-Averaged Method for Convection-Diffusion Problems on Polytopal Meshes}},
author = {James H. Adler and Xiaozhe Hu and Seulip Lee and Eoghan O'Keefe},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01175},
eprint = {2611.01175},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01175},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Incidence-Based Edge-Averaged Method for Convection-Diffusion Problems on Polytopal Meshes
AU - James H. Adler
AU - Xiaozhe Hu
AU - Seulip Lee
AU - Eoghan O'Keefe
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01175
UR - https://arxiv.org/abs/2611.01175
AB - We develop an incidence-based edge-averaged method for scalar convection-diffusion problems on convex polytopal meshes in arbitrary dimensions. The proposed method extends the classical edge-averaged finite-element scheme through a graph-theoretic formulation based on vertex-edge incidence matrices. We first establish this formulation on simplices in arbitrary dimensions and then extend it to convex polytopes by introducing complementary edges and generalized barycentric coordinates. A key ingredient is a matrix identity that relates the generalized Whitney edge mass matrix to a diagonal representation of the Poisson stiffness matrix, enabling an exponentially-fitted discretization expressed entirely in terms of nodal values and edge weights. We prove that the resulting stiffness matrix is a nonsingular M-matrix under suitable positivity and connectivity conditions, ensuring monotonicity and the discrete maximum principle. Numerical experiments on two- and three-dimensional simplicial and nonsimplicial meshes demonstrate first-order convergence in the H^1 semi-norm and stable, nonoscillatory solutions in convection-dominated regimes.
ER -