Structure-preserving Kernel-based methods for solving dissipative PDEs on surfaces

TL;DR

This work develops a meshless, structure-preserving Galerkin method for dissipative PDEs on surfaces by representing solutions in a finite-dimensional space spanned by positive definite kernels and enforcing energy dissipation through an $L^2$-projection of the variational derivative. Spatial discretization yields a semi-discrete system with a guaranteed decrease of the discrete energy, which is then integrated in time using the Average Vector Field (AVF) method to obtain a fully discrete, energy-dissipative scheme. The authors derive and analyze concrete schemes for the Allen-Cahn and Cahn-Hilliard equations, including convergence results for the Allen-Cahn equation and practical implementations via kernel-based quadrature and local Lagrange functions to achieve sparse, parallelizable computations. Extensive numerical experiments on spheres, tori, and other surfaces demonstrate high accuracy, super-convergence in space, second-order temporal accuracy, and robust energy dissipation, outperforming some mesh-free baselines. The framework provides a flexible, scalable approach for structure-preserving surface PDEs with potential applications in phase-field modeling, materials science, and geometric image analysis, with avenues for extension to additional dissipative systems on complex geometries.

Abstract

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.

Figures (10)

• Figure 1: (a) Maximum latitudinal values of the Lagrange function for the Matérn kernel $\phi_3$ and the surface spline $\psi_2$; (b) The Lagrange coefficients $\alpha_{\xi,\eta}$ constructed from the restricted Matérn kernel $\phi_3$ with the shape parameter $\epsilon=14$.
• Figure 2: The relative $L_2(\mathcal{S})$-error profiles with respect to $h_X$ for the proposed sparse kernel-based Galerkin method in Algorithm \ref{['Algor:1']}. This method is tested on the Allen-Cahn equation defined on a sphere and a torus, using a local Lagrange basis constructed by the Matérn kernel of varying smoothness orders ($m=2,3,4,5,6$). The quadrature points ($Y$) are held constant in these tests, with $N_Y=40001$ for the sphere and $N_Y=39122$ for the torus.
• Figure 3: As a counterpart to Figure \ref{['Fig.AC_L2Err_NX']}, this figure presents the relative $L_2(\mathcal{S})$-error profiles with respect to $h_Y$. Here, $N_X$ remains constant with $N_X=3721$ for the sphere and $N_X=3112$ for the torus.
• Figure 4: Numerical simulations of mean curvature flow modeled by the Allen-Cahn equation at different time levels, approximated by Algorithm \ref{['Algor:1']}. The parameters used are $\epsilon=0.05$ and $\triangle t=5\times 10^{-4}$. As time evolves, the interfaces shrink by mean curvature, indicating a decreasing radius of the spherical cap.
• Figure 5: Comparison of numerical and analytical radii demonstrating the accuracy of the proposed method (left), and scaled representation of non-increasing total discrete energy over time (right), as per Algorithm \ref{['Algor:1']} for the Allen-Cahn equation solution.

Theorems & Definitions (14)

• Lemma 2.1
• proof
• Theorem 3.1
• proof
• Theorem 4.1
• Theorem 4.2
• proof
• Lemma 4.1
• proof
• Theorem 4.3