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<p>In this paper, we propose a general meshless structure-preserving Galerkin method for solving dissipative PDEs on surfaces. By posing the PDE in the variational formulation and simulating the solution in the finite-dimensional approximation space spanned by (local) Lagrange functions generated with positive definite kernels, we obtain a semi-discrete Galerkin equation that inherits the energy dissipation property. The fully-discrete structure-preserving scheme is derived with the average vector field method. We provide a convergence analysis of the proposed method for the Allen-Cahn equation. The numerical experiments also verify the theoretical analysis including the convergence order and structure-preserving properties. Furthermore, we provide numerical evidence demonstrating that the Lagrange function and the coefficients generated by a restricted kernel decay exponentially, even though a comprehensive theory has not yet been developed.</p>

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Structure-Preserving Kernel-Based Meshless Methods for Solving Dissipative PDEs on Surfaces

  • 期刊论文

  • 发表日期：

    2025年1月27日

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  • Zhengjie Sun,
  • Leevan Ling,
  • Meng Chen

摘要

In this paper, we propose a general meshless structure-preserving Galerkin method for solving dissipative PDEs on surfaces. By posing the PDE in the variational formulation and simulating the solution in the finite-dimensional approximation space spanned by (local) Lagrange functions generated with positive definite kernels, we obtain a semi-discrete Galerkin equation that inherits the energy dissipation property. The fully-discrete structure-preserving scheme is derived with the average vector field method. We provide a convergence analysis of the proposed method for the Allen-Cahn equation. The numerical experiments also verify the theoretical analysis including the convergence order and structure-preserving properties. Furthermore, we provide numerical evidence demonstrating that the Lagrange function and the coefficients generated by a restricted kernel decay exponentially, even though a comprehensive theory has not yet been developed.