RBF-Generated Finite Difference Method Coupled with Quadratic Programming for Solving PDEs on Surfaces with Derivative Boundary Conditions

Abstract

Derivative boundary conditions introduce challenges for mesh-free discretizations of PDEs on surfaces, especially when the domain is represented by randomly sampled point clouds. The recently developed two-step tangent-space RBF-generated finite difference (RBF-FD) method provides high accuracy on closed surfaces. However, it may lose stability when applied directly to surface PDEs with derivative boundary conditions. To enhance numerical stability, we develop a mesh-free method that couples the two-step tangent-space RBF-FD discretization with a quadratic programming (QP) procedure to stabilize the operator approximation for interior points near boundaries. For boundary points, we construct restricted nearest-neighbor stencils biased in the co-normal direction and employ a constrained quadratic program to approximate outward co-normal derivatives. The resulting method avoids using ghost points and does not require quasi-uniform node distributions. We validate the approach on elliptic problems, eigenvalue problems, time-dependent diffusion equations, and elliptic interface problems on surfaces with boundary. Numerical experiments demonstrate stable performance and high-order accuracy across a variety of surfaces.

Figures (9)

• Figure 1: Poisson problems on 2D semi-torus in $\mathbb{R}^{3}$. Panels (a) and (b) show three types of interior points in intrinsic coordinates $(\psi^1,\psi^2)$ without and with quadratic programming, respectively: green dots satisfy both nearly diagonal dominance conditions, red crosses violate the first condition $w_1<0$, and blue circles violate the second condition $\gamma\geq 3$. The quadratic optimization approach completely eliminates points with $w_1 > 0$ while only partially eliminating those with $\gamma < 3$. Here no boundary points are shown and all points shown are interior. The degree is $l=4$, the number of interior points is $N_I = 6240$ and the number of boundary points is $N_B=160$.
• Figure 2: Poisson problems on 2D semi-torus in $\mathbb{R}^{3}$. Comparison of Laplacian coefficients $w_{1},\ldots,w_k$ for points away from and close to the boundary. (a) For most interior points (green dots in Fig. \ref{['fig2:torus']}), both nearly diagonal dominance conditions ($w_1<0$ and $\gamma\geq 3$) can be satisfied. However, for some points close to the boundary (red crosses in Fig. \ref{['fig2:torus']}(a)), the first condition $w_1 < 0$ may not be satisfied (panel (b)) for a certain range of $K$. After quadratic programming, $w_1 < 0$ can be satisfied for all these points (panel (c)). For other points close to the boundary (blue circles in Figs. \ref{['fig2:torus']}(a)(b)), the condition $\gamma \geq 3$ may not be met (panel (d)). After quadratic programming, some points can be improved (panel (e)) whereas some points remain to be $\gamma < 3$. The degree is $l=4$, the number of interior points is $N_I = 6240$ and the number of boundary points is $N_B=160$.
• Figure 3: Poisson problems on 2D semi-torus in $\mathbb{R}^{3}$. Panel (a) shows the pointwise absolute error of $\Delta_M$ using our combined RBF-FD and QP approach with $N=6400$ and $l=4$. Panel (b) shows the consistency of FEs for different polynomial degrees. All simulations are run with 12 independent trials, each with a set of randomly sampled data points.
• Figure 4: Stencil illustration and numerical consistency for boundary operators. Panel (a) shows the restricted KNN stencil. Panel (b) shows a typical distribution of weights obtained by the RBF-FD approach for most boundary points. Panels (c) and (d) compare the weights for the outward co-normal derivative without and with quadratic optimization, demonstrating the elimination of unstable weights. Panel (e) illustrates the pointwise forward error (FE) for a circular boundary of the semi-torus, while panel (f) shows the consistency across different polynomial degrees $l_{\mathrm{bd}}$, matching the expected theoretical convergence rate $O(N^{-l/2})$.
• Figure 5: Solution accuracy for Poisson problems on a 2D semi-torus in $\mathbb{R}^{3}$. Panels (a) and (b) display the spatial distribution of pointwise absolute errors for degrees of $(l,l_{\mathrm{bd}})=(3,2)$ and $(l,l_{\mathrm{bd}})=(4,4)$, respectively. Panels (c)--(f) illustrate the convergence of inverse errors (IEs) for $l \in \{2, 3, 4, 5\}$. All simulations are run with 12 independent trials, each with a set of randomly sampled data points.

Theorems & Definitions (2)

• Remark 2.1
• Example 2.2