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.

翻译：本文提出一种通用的无网格结构保持伽辽金方法，用于求解曲面上的耗散偏微分方程。通过将偏微分方程转化为变分形式，并在由正定核生成的（局部）拉格朗日函数所张成的有限维逼近空间中模拟解，我们获得了继承能量耗散特性的半离散伽辽金方程。全离散结构保持格式采用平均向量场方法导出。我们针对Allen-Cahn方程给出了所提方法的收敛性分析。数值实验也验证了理论分析结果，包括收敛阶数与结构保持特性。