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

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.