RBF-Generated Finite Difference Method Coupled with Quadratic Programming for Solving PDEs on Surfaces with Derivative Boundary Conditions
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911552943161344 |
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| author | Chen, Peng Jiang, Shixiao Willing Li, Rongji Yan, Qile |
| author_facet | Chen, Peng Jiang, Shixiao Willing Li, Rongji Yan, Qile |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28035 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | RBF-Generated Finite Difference Method Coupled with Quadratic Programming for Solving PDEs on Surfaces with Derivative Boundary Conditions Chen, Peng Jiang, Shixiao Willing Li, Rongji Yan, Qile Numerical Analysis 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. |
| title | RBF-Generated Finite Difference Method Coupled with Quadratic Programming for Solving PDEs on Surfaces with Derivative Boundary Conditions |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2603.28035 |