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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2511.18049 |
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| _version_ | 1866912766767398912 |
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| author | Li, Rongji Di, Haichuan Jiang, Shixiao Willing |
| author_facet | Li, Rongji Di, Haichuan Jiang, Shixiao Willing |
| contents | Solving partial differential equations (PDEs) on manifolds defined by randomly sampled point clouds is a challenging problem in scientific computing and has broad applications in various fields. In this paper, we develop a two-step generalized radial basis function-generated finite difference (gRBF-FD) method for solving PDEs on manifolds without boundaries, identified by randomly sampled point cloud data. The gRBF-FD is based on polyharmonic spline kernels and multivariate polynomials (PHS+Poly) defined over the tangent space in a local Monge coordinate system. The first step is to regress the local target function using a generalized moving least squares (GMLS) while the second step is to compensate for the residual using a PHS interpolation. Our gRBF-FD method has the same interpolant form with the standard RBF-FD but differs in interpolation coefficients. Our approach utilizes a specific weight function in both the GMLS and PHS steps and implements an automatic tuning strategy for the stencil size K (i.e., the number of nearest neighbors) at each point. These strategies are designed to produce a Laplacian matrix with a specific coefficient structure, thereby enhancing stability and reducing the solution error. We establish an error bound for the operator approximation in terms of the so-called local stencil diameter as well as in terms of the number of data. We further demonstrate the high accuracy of gRBF-FD through numerical tests on various smooth manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18049 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two-step Generalized RBF-Generated Finite Difference Method on Manifolds Li, Rongji Di, Haichuan Jiang, Shixiao Willing Numerical Analysis Solving partial differential equations (PDEs) on manifolds defined by randomly sampled point clouds is a challenging problem in scientific computing and has broad applications in various fields. In this paper, we develop a two-step generalized radial basis function-generated finite difference (gRBF-FD) method for solving PDEs on manifolds without boundaries, identified by randomly sampled point cloud data. The gRBF-FD is based on polyharmonic spline kernels and multivariate polynomials (PHS+Poly) defined over the tangent space in a local Monge coordinate system. The first step is to regress the local target function using a generalized moving least squares (GMLS) while the second step is to compensate for the residual using a PHS interpolation. Our gRBF-FD method has the same interpolant form with the standard RBF-FD but differs in interpolation coefficients. Our approach utilizes a specific weight function in both the GMLS and PHS steps and implements an automatic tuning strategy for the stencil size K (i.e., the number of nearest neighbors) at each point. These strategies are designed to produce a Laplacian matrix with a specific coefficient structure, thereby enhancing stability and reducing the solution error. We establish an error bound for the operator approximation in terms of the so-called local stencil diameter as well as in terms of the number of data. We further demonstrate the high accuracy of gRBF-FD through numerical tests on various smooth manifolds. |
| title | Two-step Generalized RBF-Generated Finite Difference Method on Manifolds |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2511.18049 |