The paper focuses on the rigorous analysis of the numerical stability of variational least-squares kernel-based methods for solving second-order elliptic partial differential equations (PDEs). It provides a formal proof for the stability inequality that was previously conjectured in the referenced work. This fills a significant theoretical gap and provides a comprehensive theoretical foundation for these methods.
The key highlights and insights are:
The paper proves the stability inequality (1.2) without using the conjecture from the referenced work. This establishes the theoretical foundations for the error estimates of the variational least-squares solution.
The paper proves another stability inequality involving weighted-discrete norms. This inequality is the key to the convergent analysis of a weighted least-squares kernel-based collocation method.
The paper compares the theoretical results of the various implementations of the kernel-based methods, providing insight into their relative efficiency and accuracy on data sets with large mesh ratios.
The paper demonstrates that the exact quadrature weights are not necessary for the weighted least-squares kernel-based collocation method to converge, as long as the weight matrix satisfies certain conditions.
Overall, the paper provides a rigorous theoretical analysis that complements the previous work and establishes a comprehensive understanding of the stability and convergence properties of these kernel-based numerical methods.
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by Meng Chen,Le... at arxiv.org 04-22-2024
https://arxiv.org/pdf/2312.07080.pdfDeeper Inquiries