A Finite Element Method for the Grad--Mercier Equation
Abstract
We develop and analyse a conforming piecewise affine finite element method for the Grad--Mercier equation, a nonlocal elliptic model for magnetic equilibria whose source depends on the global rank of a flux function. The principal difficulty is that the unique continuum equilibrium has a maximum plateau of prescribed positive area. The solution therefore samples the rank law only above the plateau mass, while perturbations that separate equal values may distribute this mass over several numerical ranks and thereby alter the non-strict rank source. We exploit the continuum well-posedness structure to replace the rank law by an equivalent clipped, nonnegative law, and replace Lebesgue measure by a mass-lumped nodal measure. Equal nodal values are assigned a common rank interval, thereby retaining the full lumped mass of each tied level. The resulting weighted rearrangement gives a strictly convex discrete energy and hence a unique finite element equilibrium. On meshes satisfying a discrete comparison principle, the discrete equilibrium is nonnegative and bounded above by a computable barrier. A Poisson-preconditioned rearrangement iteration converges on each fixed mesh and admits computable a posteriori energy and energy-norm bounds. As polygonal finite element domains converge to the physical cross-section, the discrete equilibria converge strongly in H10(Ω) and uniformly on Ω̅ to the unique Grad--Mercier equilibrium. Reconstructed equilibrium sources converge weak-* in L∞(Ω) to the original nonlocal source.
Related
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BibTeX
@misc{ashby2026finite,
title = {{A Finite Element Method for the Grad--Mercier Equation}},
author = {Ben Ashby and Tristan Pryer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05313},
eprint = {2610.05313},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05313},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Finite Element Method for the Grad--Mercier Equation
AU - Ben Ashby
AU - Tristan Pryer
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05313
UR - https://arxiv.org/abs/2610.05313
AB - We develop and analyse a conforming piecewise affine finite element method for the Grad--Mercier equation, a nonlocal elliptic model for magnetic equilibria whose source depends on the global rank of a flux function. The principal difficulty is that the unique continuum equilibrium has a maximum plateau of prescribed positive area. The solution therefore samples the rank law only above the plateau mass, while perturbations that separate equal values may distribute this mass over several numerical ranks and thereby alter the non-strict rank source. We exploit the continuum well-posedness structure to replace the rank law by an equivalent clipped, nonnegative law, and replace Lebesgue measure by a mass-lumped nodal measure. Equal nodal values are assigned a common rank interval, thereby retaining the full lumped mass of each tied level. The resulting weighted rearrangement gives a strictly convex discrete energy and hence a unique finite element equilibrium. On meshes satisfying a discrete comparison principle, the discrete equilibrium is nonnegative and bounded above by a computable barrier. A Poisson-preconditioned rearrangement iteration converges on each fixed mesh and admits computable a posteriori energy and energy-norm bounds. As polygonal finite element domains converge to the physical cross-section, the discrete equilibria converge strongly in H^1_0(Ω) and uniformly on Ω̅ to the unique Grad--Mercier equilibrium. Reconstructed equilibrium sources converge weak-* in L^∞(Ω) to the original nonlocal source.
ER -