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<p>We study superconvergent discretization of the Laplace–Beltrami operator on time-space product manifolds with Neumann temporal boundary values, which could arise for instance in the context of dynamic optimal transport on general surfaces. We propose a coupled scheme that combines finite difference methods in time with surface finite element methods in space. By establishing a new summation by parts formula and proving the supercloseness of the semi-discrete solution, we derive superconvergence results for the recovered gradient via post-processing techniques. In addition, our geometric error analysis is implemented within a novel framework based on the approximation of the Riemannian metric. Several numerical examples are provided to validate and illustrate the theoretical results.</p>

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An FDM-sFEM Scheme on Time-Space Manifolds and Its Superconvergence Analysis

  • 期刊论文

  • 发表日期：

    2026年3月19日

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  • Chengrun Jiang,
  • Guozhi Dong,
  • Hailong Guo,
  • Zuoqiang Shi

摘要

We study superconvergent discretization of the Laplace–Beltrami operator on time-space product manifolds with Neumann temporal boundary values, which could arise for instance in the context of dynamic optimal transport on general surfaces. We propose a coupled scheme that combines finite difference methods in time with surface finite element methods in space. By establishing a new summation by parts formula and proving the supercloseness of the semi-discrete solution, we derive superconvergence results for the recovered gradient via post-processing techniques. In addition, our geometric error analysis is implemented within a novel framework based on the approximation of the Riemannian metric. Several numerical examples are provided to validate and illustrate the theoretical results.