Energy Stability and Error Estimates for a Globally Divergence-Free DG Method of Arbitrary Order
Abstract
We propose and analyze an arbitrary-order globally divergence-free (GDF) discontinuous Galerkin (DG) method for the two-dimensional linear magnetic induction equation on unstructured triangular meshes. To the best of our knowledge, this is the first work to provide a rigorous energy-based analysis of high-order GDF-DG methods. First, we incorporate the edge normal-trace evolution into a closed, strictly local semidiscrete operator acting on the elementwise magnetic field alone. Then, we introduce two local modifications to the continuous electric field: a weighted patch moment correction that enforces local polynomial orthogonality and a vertex-patch stabilization that damps the remaining high-order modes. On shape-regular, quasi-uniform periodic triangulations, we prove semidiscrete energy stability with a sufficiently large mesh-independent penalty and establish a priori error estimates for smooth solutions. Numerical experiments support the analysis.
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BibTeX
@misc{liu2026energy,
title = {{Energy Stability and Error Estimates for a Globally Divergence-Free DG Method of Arbitrary Order}},
author = {Yuchang Liu and Yan Jiang and Mengping Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39299},
eprint = {2610.39299},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39299},
}
RIS
TY - UNPB
TI - Energy Stability and Error Estimates for a Globally Divergence-Free DG Method of Arbitrary Order
AU - Yuchang Liu
AU - Yan Jiang
AU - Mengping Zhang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39299
UR - https://arxiv.org/abs/2610.39299
AB - We propose and analyze an arbitrary-order globally divergence-free (GDF) discontinuous Galerkin (DG) method for the two-dimensional linear magnetic induction equation on unstructured triangular meshes. To the best of our knowledge, this is the first work to provide a rigorous energy-based analysis of high-order GDF-DG methods. First, we incorporate the edge normal-trace evolution into a closed, strictly local semidiscrete operator acting on the elementwise magnetic field alone. Then, we introduce two local modifications to the continuous electric field: a weighted patch moment correction that enforces local polynomial orthogonality and a vertex-patch stabilization that damps the remaining high-order modes. On shape-regular, quasi-uniform periodic triangulations, we prove semidiscrete energy stability with a sufficiently large mesh-independent penalty and establish a priori error estimates for smooth solutions. Numerical experiments support the analysis.
ER -