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Kernel-based meshfree collocation method for solving nonlinear and parametric PDEs
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DOI：10.1016/j.apnum.2026.01.018.png)
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Abstract
En 中文
In this paper, we develop a new intrusive numerical method (kernel-based meshfree collocation method) for solving nonlinear and parametric PDEs. Our method only consists of two important steps - trial and testing, and is easy to implement. The new method can avoid the tedious mesh generation and the domain integration and is capable of dealing with any irregular distribution of nodes. A general framework for proving the convergence of meshfree collocation method for solving well-posed nonlinear and parametric PDEs is presented. In terms of kernel trial, the optional approximation spaces might include a large class of finite dimensional approximation spaces such as radial basis functions spaces, spline functions spaces, local Lagrange function spaces, spectral techniques, finite element spaces and so on. In terms of testing, it only requires taking values directly on the collocation points and greatly simplifies the difficulties caused by variation and integration. The general theoretical results cover the error bounds and convergence rates. In order to show how the general theoretical framework can be set to work, we take a specific model as an illustration and derive specific convergence rates in Sobolev spaces. The convergence rates depend on the regularity of the exact solution, the smoothness of the computing domain (combination of physical domain and parametric domain), the sampling inequality, the inverse inequality, the approximation of radial basis functions trial spaces, and the effectiveness of algebraic solvers. Several numerical examples are provided to illustrate the accuracy and demonstrate numerical efficiency.
Keywords:
Parametric PDEs
Collocation methods
Radial basis functions
Meshfree methods
Convergence rates