A Meshfree RBF-FD Constant along Normal Method for Solving PDEs on Surfaces

TL;DR

The paper develops a meshfree, constant-along-normal RBF-FD method with polyharmonic spline kernels augmented by polynomials (PHS+poly) to solve PDEs on surfaces embedded in $\mathbb{R}^3$. By extending the stencil only along the normal at a single reference node, the approach overcomes polynomial rank deficiency without requiring a closest-point map, and uses local embedded stencils to compute surface derivatives. It demonstrates high-order convergence for diffusion, advection, and Turing patterns on implicit surfaces and point clouds, and extends naturally to moving surfaces via a simple particle-coupled framework. The method shows stability (negative real parts in the Laplace–Beltrami spectrum), robustness across high-curvature geometries, and practical applicability to evolving geometries and complex topologies.

Abstract

This paper introduces a novel meshfree methodology based on Radial Basis Function-Finite Difference (RBF-FD) approximations for the numerical solution of partial differential equations (PDEs) on surfaces of codimension 1 embedded in $\mathbb{R}^3$. The method is built upon the principles of the closest point method, without the use of a grid or a closest point mapping. We show that the combination of local embedded stencils with these principles can be employed to approximate surface derivatives using polyharmonic spline kernels and polynomials (PHS+Poly) RBF-FD. Specifically, we show that it is enough to consider a constant extension along the normal direction only at a single node to overcome the rank deficiency of the polynomial basis. An extensive parameter analysis is presented to test the dependence of the approach. We demonstrate high-order convergence rates on problems involving surface advection and surface diffusion, and solve Turing pattern formations on surfaces defined either implicitly or by point clouds. Moreover, a simple coupling approach with a particle tracking method demonstrates the potential of the proposed method in solving PDEs on evolving surfaces in the normal direction. Our numerical results confirm the stability, flexibility, and high-order algebraic convergence of the approach.

Figures (14)

• Figure 1: A visualization of our method for calculating RBF-FD weights at a reference node $\boldsymbol{x}_i$ (in red). The introduced stencils use $n_s$ surface nodes (within the dark gray area) and $n_{\perp}$ out-of-surface nodes along the normal direction $\vec{n}_i$ to the reference node $\boldsymbol{x}_i$, with a distance of $\varepsilon h$ between each node. Note the exaggerated spacing between nodes along the normal direction in this figure (as $0<\varepsilon < 1$). The weights of the total $n$ nodes ($\boldsymbol{w}_{s}$ for the on-surface nodes and $\boldsymbol{w}_{\perp}$ for the out-of-surface nodes) in the stencil are shown as spheres, with blue indicating negative values and yellow indicating positive values. The radius of the spheres corresponds to the square root of the magnitude of the weights. Using the closest point principles, the final stencil ($\hat{\boldsymbol{w}}_s$) consists of only $n_s$ on-surface weights.
• Figure 1: The convergence of the method when solving \ref{['forcedHeatPDE']} on two different surfaces in $\mathbb{R}^3$ with $n_\perp=14$ and $\varepsilon=0.2$.
• Figure 1: Snapshots of the solution of the cross-diffusion reaction-diffusion system on the unit sphere (top) and the Dziuk's surface (bottom) at times $t=0.25$, $0.5$ and $0.75$.
• Figure 1: Eigenvalue spectra of $\Delta_{\Gamma}$ for the tooth and unit sphere from Section \ref{['sec_param3Dcase']} with $N = 20{,}298$ ($h = 0.036$) and $N = 10{,}000$ ($h = 0.035$) nodes, respectively, when using RBF-FD $r^5$ augmented with polynomials of degree $l=2,4$ and $6$.
• Figure 2: Effect of the constant extension along the normal when approximating the surface Laplacian of $u(x,y) = \sin(\pi x)\cos(\pi y)$ on the unit circle $\Gamma$ at $(0.9980, 0.0628)$ with $n_s = 20$ using PHS $r^7$ with polynomials of degree 2: (a) Contour lines vs. $n_{\perp}$ for $\varepsilon=1$; (b) Contour lines vs. $\varepsilon$ for $n_{\perp}=8$; (c) Effect of varying $\varepsilon$ and $n_{\perp}$ on the residual of the surface Laplacian, projection of surface gradient on normal direction and condition number of the collocation system.