  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.
