High-order energy-stable BGN parametric finite element methods for geometric flows
Abstract
We construct high-order Runge--Kutta extensions of Barrett--Garcke--Nürnberg (BGN) parametric finite element methods for curve-shortening flow and curve diffusion of planar curves and for mean curvature flow and surface diffusion of closed genus-$0$ surfaces. On each time slab Im=(tm,tm+1], the continuous equations are posed on the left-endpoint surface Γm=Γtm through the map Xm,t:Γm→Γt, whose target is the evolving surface at time t. For curves, this formulation follows by harmonic pullback from Γ0 to Γm. For surfaces, an orientation-preserving harmonic diffeomorphism is posed separately on each time slab, and conformality yields the weight 1/2|∇ΓmXm,t|2. Evaluating the time-slab equations at the Runge--Kutta internal times and applying mass-lumped parametric finite elements yields systems on the common domain Γm, with Γtm+ciτm as the intermediate target geometry. This internal-time discretization constructs high-order BGN-structured extensions for the four flows. For an algebraically stable tableau with nonnegative weights, every exact stage solution with nondegenerate intermediate configurations satisfies monotone decay of the discrete curve length or surface area for every positive time step for which that solution exists. Radau IIA experiments exhibit energy decay for all four flows and BGN-type mesh redistribution in the curve tests. The Hausdorff self-convergence results exhibit high-order behavior consistent with the corresponding design orders.
Related
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- High-order energy diminishing ALE-SAV finite element methods for two-phase Navier--Stokes flow
BibTeX
@misc{ma2026highorder,
title = {{High-order energy-stable BGN parametric finite element methods for geometric flows}},
author = {Shu Ma and Qiqi Rao},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29877},
eprint = {2608.29877},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29877},
}
RIS
TY - UNPB
TI - High-order energy-stable BGN parametric finite element methods for geometric flows
AU - Shu Ma
AU - Qiqi Rao
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29877
UR - https://arxiv.org/abs/2608.29877
AB - We construct high-order Runge--Kutta extensions of Barrett--Garcke--Nürnberg (BGN) parametric finite element methods for curve-shortening flow and curve diffusion of planar curves and for mean curvature flow and surface diffusion of closed genus-$0$ surfaces. On each time slab I_m=(t_m,t_m+1], the continuous equations are posed on the left-endpoint surface Γ^m=Γ^t_m through the map X^m,t:Γ^m→Γ^t, whose target is the evolving surface at time t. For curves, this formulation follows by harmonic pullback from Γ^0 to Γ^m. For surfaces, an orientation-preserving harmonic diffeomorphism is posed separately on each time slab, and conformality yields the weight 1/2|∇_Γ^mX^m,t|^2. Evaluating the time-slab equations at the Runge--Kutta internal times and applying mass-lumped parametric finite elements yields systems on the common domain Γ^m, with Γ^t_m+c_iτ_m as the intermediate target geometry. This internal-time discretization constructs high-order BGN-structured extensions for the four flows. For an algebraically stable tableau with nonnegative weights, every exact stage solution with nondegenerate intermediate configurations satisfies monotone decay of the discrete curve length or surface area for every positive time step for which that solution exists. Radau IIA experiments exhibit energy decay for all four flows and BGN-type mesh redistribution in the curve tests. The Hausdorff self-convergence results exhibit high-order behavior consistent with the corresponding design orders.
ER -