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High-order energy-stable BGN parametric finite element methods for geometric flows
Shu Ma, Qiqi Rao
TL;DR
The paper addresses how to extend BGN parametric finite element methods to high-order time discretizations for four geometric flows while retaining their energy-stable structure. It uses common-domain harmonic time-slab formulations with implicit Runge–Kutta stages and mass-lumped finite elements. Algebraically stable methods yield monotone discrete length or area decay under nondegenerate exact stage solutions, while Radau IIA experiments show the predicted energy, mesh, and convergence behavior.
Problem
The paper targets high-order discretizations of BGN methods for curve and surface geometric flows while retaining their paired structure and energy stability.
Method
The method uses harmonic or conformal common-domain time-slab formulations, evaluates them at implicit Runge–Kutta internal times, and discretizes them with mass-lumped parametric finite elements.
Results
Algebraically stable Runge–Kutta methods produce monotone discrete curve-length or surface-area decay for exact nondegenerate stage solutions, with Radau IIA experiments confirming energy decay for all four flows.
Takeaways & Limitations
The construction retains BGN-type curve mesh redistribution and exhibits Hausdorff self-convergence consistent with the corresponding design orders.
Takeaways & Limitations
The formulation assumes orientation-preserving harmonic maps with specified smoothness and nondegeneracy conditions on the evolving configurations.
Abstract
from arXiv · showhide
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 $\X^{m,t}:Γ^m\toΓ^t$, 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 $\frac12|\nabla_{Γ^m}\X^{m,t}|^2$. 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.
1 Introduction
The paper develops high-order Runge–Kutta extensions of BGN parametric finite element methods for four geometric flows using common-domain time-slab formulations. The resulting schemes preserve BGN structure and support unconditional discrete energy decay under stated stage-solution conditions.
- A common-domain formulation places each time slab on the left-endpoint geometry Γm while mapping to the evolving target Γt.Curves use harmonic pullback, whereas surfaces use an intervalwise orientation-preserving harmonic diffeomorphism.
- The four target flows are curve-shortening flow, curve diffusion, mean curvature flow, and surface diffusion for planar curves and closed genus-0 surfaces.
- The internal-time discretization evaluates velocity–curvature pairs at Runge–Kutta stages and applies mass-lumped parametric finite elements on Γm.This retains the paired BGN structure at high order in time.
- Every positive time step admits discrete length or area decay for algebraically stable methods with nonnegative weights whenever an exact nondegenerate stage solution exists.The discrete energy is the polygonal curve length or polyhedral surface area.
- Radau IIA experiments show monotone energy decay for all four flows, BGN-type curve mesh redistribution, and high-order Hausdorff self-convergence consistent with design orders.
2 PFEM and Runge–Kutta methods
The PFEM framework evolves polygonal or polyhedral geometry through nodal velocities and a discrete flow map, while implicit Runge–Kutta methods provide stage relations and an algebraic identity for energy analysis.
- 2.1 Parametric finite element method: Parametric finite elements represent evolving interfaces directly by polygonal curves or triangulated surfaces determined by nodal positions and connectivity.
- 2.1 Parametric finite element method: The nodal vector evolves according to the interface velocity, and its piecewise linear interpolant defines the discrete flow map to the evolving mesh.
- 2.1 Parametric finite element method: Mass-lumped inner products use elementwise vertex quadrature on polygonal edges or triangular surface elements.
- 2.2 Runge–Kutta methods and algebraic stability: An s-stage implicit Runge–Kutta method advances the semidiscrete system through stage values and a weighted update defined by its Butcher tableau.
- 2.2 Runge–Kutta methods and algebraic stability: Algebraic stability requires nonnegative weights and a positive-semidefinite matrix, including the Radau IIA and Lobatto IIIC families.
- 2.2 Runge–Kutta methods and algebraic stability: The algebraic Runge–Kutta identity yields norm decay when every stage satisfies a nonpositive inner-product condition.The proof combines the nonpositive stage contribution with a nonnegative Gram-matrix contribution.
3 RK-based PFEMs for curve flows
The curve-flow schemes pull harmonic-map formulations back to the left-endpoint curve, placing all Runge–Kutta stages on a common domain while retaining the BGN structure. Algebraic stability then yields discrete length decay for curve-shortening flow and curve diffusion under nondegeneracy assumptions.
- Time-slab formulation: The harmonic-map system is transported from Γ^0 to Γ^m, where internal Runge–Kutta times expose a metric-weighted BGN pair on the common domain Γ^m.For curve-shortening flow, the normal-velocity relation is multiplied by the metric weight |∇ΓmXm,t|^2, producing the paired formulation used for discretization.
- Time-slab formulation: The pullback preserves harmonic structure between time-node curves and yields identities for transported functions, including the Laplace–Beltrami operator.The construction relies on harmonic parametrizations, constant-speed properties, and pullback maps Xr,t between intermediate curves.
- Finite element discretization: Mass-lumped parametric finite elements discretize zero-order products, while standard elementwise integration treats gradients and fixed-point iterations freeze metric weights and intermediate normals.The resulting nonlinear stage problems reduce during each fixed-point iteration to sparse linear BGN systems on the common finite element space.
- Curve-shortening flow: Algebraically stable Runge–Kutta methods give time-step-unrestricted length decay for curve-shortening flow when the exact nonlinear stage solution has nondegenerate intermediate polygonal curves.The proof combines per-stage energy decay with the Runge–Kutta algebraic identity and the nonpositive Gram-matrix contribution.
- Curve diffusion: The same construction gives energy decay for curve diffusion through a metric-weighted curvature equation and a per-stage estimate combined with a geometric inequality.The theorem applies for every positive time step under exact solvability and nondegeneracy of the intermediate polygonal curves.
4 RK-based PFEMs for surface flows
For closed genus-0 surfaces, the paper formulates mean curvature flow and surface diffusion on each time slab’s left-endpoint geometry, then discretizes internal Runge–Kutta times with mass-lumped parametric finite elements. Algebraic stability and stagewise BGN cancellations yield discrete surface-area decay under nondegeneracy assumptions.
- Time-slab formulation: The surface formulation uses an orientation-preserving harmonic diffeomorphism from the left-endpoint surface Γm to each intermediate target Γt.Conformality of this map supplies the weighted mean-curvature relation used in the surface-flow formulation.
- Time-slab formulation: The normal-velocity equation is weighted by |∇ΓmXm,t|^2, producing the formulation discretized by the Runge–Kutta stages.The weight enters both the normal-velocity relation and the weak formulation on the common domain Γm.
- Energy stability: Mean curvature flow satisfies per-stage energy decay, and algebraic stability converts this stage estimate into decay for the fully discrete method.The theorem assumes an exact nonlinear stage solution with nondegenerate intermediate triangulated surfaces and allows any positive time step.
- Runge–Kutta discretization: Evaluating the weak system at Runge–Kutta internal times and applying mass-lumped parametric finite elements gives stage equations on Γm.The intermediate target geometry is represented through the stage maps and pulled-back normals, followed by a common endpoint update and mesh push-forward.
- Energy stability: Surface diffusion has the same stagewise cancellation structure, yielding discrete surface-area decay for algebraically stable Runge–Kutta methods.The result applies when the nonlinear stage system has an exact solution whose intermediate triangulated surfaces are nondegenerate.
5 Numerical results
Radau IIA experiments test temporal convergence, energy decay, and mesh quality for the four high-order BGN flows. The observed self-convergence matches design orders, energies decay, and curve meshes become more uniform than with classical BGN.
- Experimental setup: Radau IIA experiments with s = 1, 2, 3 examine temporal convergence, discrete energy decay, and mesh quality for curves and closed genus-0 surfaces.The computations use the Radau IIA family, whose classical order is k = 2s −1.
- Temporal convergence: The fitted slopes for regular polygon, icosahedron, and asymmetric-curve tests agree with the theoretical convergence orders.The curve and surface tests use Hausdorff self-errors, with O(τ 2s−1) reference behavior.
- Flower-curve flows: The flower-curve curve-shortening computation reduces discrete length from approximately 19.72 to 4.83 while the mesh-quality ratio changes from approximately 13 to 1.15 for Radau and 1.22 for BGN.The curve-shortening test uses s = 2, 3, 800 time steps, and 112 vertices.
- Flower-curve flows: The flower-curve diffusion computation reduces length from approximately 19.72 to 6.85 and the mesh-quality ratio from approximately 13 to 1.08 for Radau and 1.27 for BGN.After initial redistribution, the Radau curves lie below the BGN reference in both mesh-quality plots.
- Surface flows of a cube: For cube mean curvature flow, final areas are 8.7390, 8.7334, and 8.7333 for s = 1, 2, 3; for surface diffusion, they are 19.4515, 19.3830, and 19.3787.The area histories are monotone in the reported computations.
6 Conclusion
The conclusion identifies a common-domain, internal-time finite element construction that extends the paired BGN structure to high-order Runge–Kutta methods. Algebraic stability yields one-step decay of discrete length or area, while Radau IIA tests show decay, curve mesh redistribution, and design-order self-convergence.
- Method: Internal-time evaluation on the left-endpoint geometry and mass-lumped parametric finite elements produce high-order BGN extensions for all four flows.For surfaces, conformality identifies the metric density in the common-domain formulation.
- Energy stability: Algebraic stability with nonnegative weights and nondegenerate exact stage solutions gives one-step decay of discrete curve length or surface area for positive time steps.The decay follows from stagewise BGN energy decay and the algebraic Runge–Kutta identity.
- Numerical evidence: Radau IIA experiments exhibit energy decay for all four flows, BGN-type mesh redistribution for curves, and high-order self-convergence consistent with the design orders.The conclusion ties the numerical observations to the theoretical stability and convergence structure.