An efficient radial basis function-finite difference method for high-order partial differential equations on surfaces

In this paper, we develop the radial basis function-finite difference (RBF-FD) method for solving high-order partial differential equations (PDEs) on surfaces. The fourth-order and sixth-order Laplace–Beltrami operators are approximated by the RBF-FD method augmented with multivariate polynomials. Additionally, the proposed method is utilized for solving biharmonic and triharmonic problems posed on surfaces. As a local meshfree method, the resulting differentiation matrix for the Laplace–Beltrami operator is sparse. Consequently, the sparse linear algebraic system can be rapidly solved using Krylov subspace iterative method. We analyze the computational complexity of the RBF-FD method for solving high-order PDEs on surfaces. Numerical results demonstrate that our method achieves high accuracy and high convergence order. For applications, implicit time discretization is employed to simulate high-order phase-field equations on surfaces.