High-Order Invariant-Domain Preserving Continuous Finite Elements via Graph-Poisson Convex Limiting
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-ℙ1 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 ℙk lumped masses (e.g., ℙ1, ℙ3, ℙ5 on triangles), convex limiting of these fluxes conserves the ℙk mass exactly and in local flux form. For degrees with zero or negative lumped masses (e.g., ℙ2, ℙ4), 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.
Related
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BibTeX
@misc{nazarov2026highorder,
title = {{High-Order Invariant-Domain Preserving Continuous Finite Elements via Graph-Poisson Convex Limiting}},
author = {Murtazo Nazarov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.23397},
eprint = {2606.23397},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.23397},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - High-Order Invariant-Domain Preserving Continuous Finite Elements via Graph-Poisson Convex Limiting
AU - Murtazo Nazarov
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2606.23397
UR - https://arxiv.org/abs/2606.23397
AB - 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-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 ℙ_k lumped masses (e.g., ℙ_1, ℙ_3, ℙ_5 on triangles), convex limiting of these fluxes conserves the ℙ_k mass exactly and in local flux form. For degrees with zero or negative lumped masses (e.g., ℙ_2, 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.
ER -