A revisit to DeTurck method on curve shortening flow with optimal error analysis

Abstract: The curve shortening--DeTurck flow introduced by Elliott and Fritz (IMA J. Numer. Anal. 37(2): 543--603, 2017) is a reparameterization of the curve shortening flow via the DeTurck trick. For corresponding discrete schemes, although the $H^1$-error estimate has been established, the optimal $L^2$-error estimate remains open due to technical difficulties. In this paper, we prove optimal $L^2$-error estimates for linearized Euler and Crank--Nicolson discretizations combined with finite elements of degree $k\ge 1$ in space. Moreover, we provide an extrinsic approach to derive the curve shortening--DeTurck flow. Numerical experiments are presented to illustrate the mesh distribution properties and to verify the convergence rates in both space and time.

• The paper establishes optimal L2 error bounds for fully discrete DeTurck schemes using linearized Euler and Crank–Nicolson methods.
• The approach couples geometric evolution with harmonic map heat flow to ensure near-uniform mesh distribution and superconvergence in the H1 seminorm.
• Numerical experiments confirm robust convergence rates, validating the theoretical framework and mesh-preserving properties of the discretizations.

Optimal $L^2$-Error Analysis of Fully Discrete DeTurck Methods for Curve Shortening Flow

Introduction and Motivation

The paper addresses the numerical analysis of curve shortening flows (CSF), focusing on optimal $L^2$-error estimates for fully discrete DeTurck methods. The DeTurck trick, introduced in geometric analysis, reparameterizes parabolic flows (such as CSF and mean curvature flow) to incorporate tangential velocities, thus leading to better mesh quality in practical discretizations. While the $H^1$-error analysis for CSF discretizations has been well-established, optimal $L^2$-error bounds—particularly for fully discrete schemes with lower-order ($k\geq1$) polynomial finite elements—remained unresolved due to significant technical challenges. The central contribution is the proof of such optimal error bounds for both linearized Euler and Crank–Nicolson time-stepping schemes.

Revisiting the DeTurck Method for CSF

The DeTurck method modifies the standard CSF by introducing a tangential velocity component through a reparameterization inspired by the harmonic map heat flow. The paper rigorously derives this approach, demonstrating that the DeTurck-modified flow can be viewed as a coupling of the geometric evolution equation with a harmonic map heat flow from a reference manifold (curve) to the evolving curve. This ensures that mesh points remain well-distributed under evolution, combating the degeneracy issues that arise in pure normal-motion flow (Dziuk's scheme).

The formulation recovers as special cases several prominent schemes in the literature. Specifically, Elliott and Fritz's DeTurck-modified flow and the Deckelnick–Dziuk tangential redistribution method are both shown to result from particular choices of the coupling diffusion coefficient in the general harmonic map framework. The theoretical underpinning connects the reduction of perimeter (length functional) and harmonic (Dirichlet) energy in the evolving curve, justifying the favorable mesh behavior observed in practice.

Fully Discrete Schemes and Main Error Estimates

The paper constructs fully discrete schemes via finite elements of degree $k\geq1$ in space and applies two classes of time discretization: linearized Euler and Crank–Nicolson. A rigorous $L^2$-error analysis is developed for both, leveraging superconvergence results for the $H^1$-seminorm of the error. The error decomposition employs the Ritz projection (or, for $k=1$, interpolation), and the estimates crucially depend on demonstrating that the time and spatial discretization errors can be controlled optimally.

An overview of the key results:

• Linearized Euler Scheme: Error bounds established at order $\mathcal{O}(\Delta t + h^{k+1})$ in the $L^2$0 norm and $L^2$1 in the $L^2$2 seminorm.
• Linearized Crank–Nicolson Scheme: Higher-order accuracy, with $L^2$3 errors at $L^2$4 and $L^2$5 errors at $L^2$6.

These results are encapsulated in theorems with detailed proofs, using discrete Gronwall inequalities and careful treatment of nonlinearities in the geometric evolution.

Numerical Verification and Mesh Distribution Analysis

The theoretical analysis is substantiated via comprehensive numerical experiments. Both convergence in space and convergence in time are verified for the Crank–Nicolson and Euler schemes, with numerical rates matching the predicted orders for $L^2$7 finite elements. The experiments also highlight the superior mesh equidistribution properties of the DeTurck-based schemes compared to those using only normal velocity.

Mesh quality is quantified both visually and via an index comparing the ratios of curved edge lengths between the evolving and reference meshes. The mesh index rapidly approaches 1 during evolution, confirming that the DeTurck modification yields near-uniform mesh distributions, even in the presence of complex curve geometries and non-uniform initial parameterizations.

Energy dissipation properties (both curve length and Dirichlet energy) are monitored, confirming the monotonic decrease expected from the analytical reformulation.

Implications and Theoretical Impact

The main implication of this work is the rigorous establishment of optimal $L^2$8-error bounds for DeTurck-based CSF discretizations with $L^2$9 finite elements and standard time stepping. This closes a significant theoretical gap in the numerical analysis of geometric flows, where $H^1$0 estimates are essential for understanding the approximation of geometry itself, not merely its tangents or normals.

Practically, the results support the use of DeTurck-inspired methods for robust, reliable simulation of evolving curves, especially in contexts requiring adaptive or high-quality mesh evolution. The harmonic map coupling perspective also offers a systematic framework for mesh-preserving discretizations in higher-dimensional and more complex geometric flows, suggesting avenues for design of adaptive algorithms.

Theoretically, the connection of Dirichlet energy reduction with mesh equidistribution generalizes beyond CSF, impacting the analysis and development of structure-preserving geometric integrators for surfaces and higher-codimension manifolds. These results also suggest a principled approach to coupling geometric evolution with auxiliary flows (harmonic map heat flow, conformal map flows) to control mesh quality and error evolution.

Future Directions

Building on the robust $H^1$1-error framework, future research directions include:

• Extension to higher-codimension flows (e.g., surface evolution in $H^1$2)
• Optimal error analysis for anisotropic and surface flows, using the derived coupling framework
• Development and analysis of adaptive mesh refinement strategies based on harmonic map coupling
• Exploration of DeTurck-type modifications for other geometric variational flows (surface diffusion, Willmore flow)
• Investigation of convergence and error propagation in long-time or singularity-developing evolutions

Conclusion

This work rigorously proves optimal $H^1$3-error estimates for linearized Euler and Crank–Nicolson finite element discretizations of the curve shortening–DeTurck flow for $H^1$4. By establishing superconvergence in the $H^1$5-seminorm and harnessing harmonic map heat flow insights, the paper provides both theoretical guarantees and practical algorithms for mesh-preserving geometric evolution. The results not only close a longstanding gap in numerical geometric analysis but also provide a foundation for further advanced studies in the numerical approximation of geometric partial differential equations.

Reference: "A revisit to DeTurck method on curve shortening flow with optimal error analysis" (2607.04105)