RBF-based partition of unity methods for two-dimensional time-dependent PDEs: Numerical and theoretical aspects

This article discusses standard Radial Basis Function (RBF) partition of unity method and an enhanced version known as the direct RBF partition of unity method. We introduce the framework of these methods for the effective and accurate numerical analysis of certain two-dimensional time-dependent Partial Differential Equations (PDEs). Additionally, we conduct a convergence analysis of these methods and establish error bounds for local approximations. These error bounds are determined based on conditions associated with the eigenvalues of the Laplacian operator matrices generated by the proposed methods. To validate the accuracy and reliability of these approaches, we provide several numerical examples. We assess the consistency between theoretical and numerical results. The localized techniques introduced in this research demonstrate a significant reduction in computational cost and algorithmic complexity, achieved through the direct local procedure. Despite this computational advantage, both versions of the RBF partition of unity method exhibit a similar level of accuracy. Finally, we conduct a comparative analysis between the RBF partition of unity local methods and the RBF-generated finite differences method, offering valuable insights into the strengths and limitations of each approach.