High-Order Invariant-Domain Preserving Continuous Finite Elements via Graph-Poisson Convex Limiting
Year: 2026
Type: preprint
Abstract: We develop a high-order, locally conservative, invariant-domain preserving continuous finite element method for nonlinear scalar conservation laws. A first-order graph-viscosity scheme is built on the fine-$\mathbb P_1$ submesh induced by the high-order nodes, so that it remains compact and has positive lumped masses for every polynomial degree, while the high-order candidate can be any stabilized Galerkin method; here we use residual-based artificial viscosity and continuous interior penalty methods. The two updates conserve different discrete masses, and comparing each in its own conserved measure gives a nodal difference with zero sum, which a graph-Poisson problem represents as antisymmetric edge fluxes. For polynomial degrees with positive $\mathbb P_k$ lumped masses (e.g., $\mathbb P_1$, $\mathbb P_3$, $\mathbb P_5$ on triangles), convex limiting of these fluxes conserves the $\mathbb P_k$ mass exactly and in local flux form. For degrees with zero or negative lumped masses (e.g., $\mathbb P_2$, $\mathbb P_4$), limiting perturbs the mass only on the elements where it acts, and this perturbation is removed inside each such element without violating the bounds, exactly whenever the element has enough room within its bounds. Numerical experiments demonstrate high-order accuracy for smooth solutions, robustness near discontinuities, and agreement with the entropy solution on standard benchmarks.
Source: arXiv (Cornell University)
Author: Nazarov, Murtazo
Institution: Uppsala University
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Funder: Vetenskapsrådet
OpenAlex ID: W7165620691