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<p>We propose a high-order-in-time parametric finite element method (PFEM) for the planar Willmore flow, where both first- and second-order fully discrete schemes preserve energy stability. To ensure unconditional energy stability and high-order temporal accuracy, our formulation embeds a scalar Lagrange multiplier and an associated energy-related evolution equation into the GNZ formulation (Garcke, Nürnberg and Zhao, 2025&#xa0;[<CitationRef CitationID="CR16">16</CitationRef>]). This approach inherits the advantages of the GNZ framework–specifically, favorable mesh quality and the ability to simulate complex initial curve evolutions–while extending energy stability to second-order schemes and achieving high-order temporal accuracy. Numerical experiments validate the expected temporal accuracy and demonstrate the preservation of geometric properties and mesh quality throughout the evolution.</p>

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An Energy-Stable Parametric Finite Element Framework with Lagrange Multiplier for the Planar Willmore Flow

  • 期刊论文

  • 发表日期：

    2026年7月16日

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  • Zhiqing Pan,
  • Jiwei Jia

摘要

We propose a high-order-in-time parametric finite element method (PFEM) for the planar Willmore flow, where both first- and second-order fully discrete schemes preserve energy stability. To ensure unconditional energy stability and high-order temporal accuracy, our formulation embeds a scalar Lagrange multiplier and an associated energy-related evolution equation into the GNZ formulation (Garcke, Nürnberg and Zhao, 2025 [16]). This approach inherits the advantages of the GNZ framework–specifically, favorable mesh quality and the ability to simulate complex initial curve evolutions–while extending energy stability to second-order schemes and achieving high-order temporal accuracy. Numerical experiments validate the expected temporal accuracy and demonstrate the preservation of geometric properties and mesh quality throughout the evolution.