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arXiv · 2606.23397

High-Order Invariant-Domain Preserving Continuous Finite Elements via Graph-Poisson Convex Limiting

Murtazo Nazarov
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Abstract

We develop a high-order invariant-domain preserving continuous finite element method for nonlinear scalar conservation laws. The method combines a residual-viscosity high-order discretization with a low-order invariant-domain scheme constructed on a fine $\mathbb P_1$ submesh induced by the high-order nodal points. This separation avoids the restrictions caused by nonpositive lumped masses and overly wide high-order graph stencils. The high- and low-order updates are connected by a graph-Poisson flux reconstruction, which represents their difference as conservative antisymmetric graph fluxes. These fluxes are limited using convex limiting coefficients, followed by a capacity-based mass redistribution step that restores conservation while preserving the prescribed bounds whenever sufficient admissible capacity is available. Numerical experiments for smooth and nonsmooth scalar conservation laws demonstrate high-order accuracy in smooth regimes, robustness near discontinuities, and convergence to the entropy solution for challenging benchmarks.

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Murtazo Nazarov. 2026-06-22. High-Order Invariant-Domain Preserving Continuous Finite Elements via Graph-Poisson Convex Limiting. https://arxiv.org/abs/2606.23397

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