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.