In this talk, we present a Direct Radial Basis Function Partition of Unity (D-RBF-PU) method for solving partial differential equations (PDEs), which utilizes shape-parameter-dependent kernels to enhance both flexibility and accuracy [1]. Unlike traditional polyharmonic splines, our approach incorporates local radial kernels with shape parameters and polynomial terms, offering superior approximation capa- bilities. The D-RBF-PU method [2] simplifies the standard RBF-PU approach by directly discretising the solution, eliminating the need for derivative computations of partition of unity weights. This significantly reduces computational costs and improves efficiency, especially for large-scale problems. We compare our method to the RBF finite difference (RBF-FD) technique [3] and demonstrate that D-RBF-PU achieves similar accuracy while being computationally more efficient. Numerical results for elliptic PDEs, such as the Poisson equation, show rapid error decay, with ∞-norm errors as low as 10^−6 for large datasets. The method’s sparse matrix assembly and O(N^ ̄ 3) complexity make it ideal for large-scale simulations. Finally, we discuss future work, including automated shape-parameter selection and extensions to time-dependent problems. REFERENCES [1] R. Cavoretto, A. De Rossi, A. Haider, A shape-parameterized RBF-partition of unity technique for PDEs, Appl. Math. Lett., 163, 109453 (2025). [2] D. Mirzaei, The direct radial basis function partition of unity (D-RBF-PU) method for solving PDEs, SIAM J. Sci. Comput. 43, A54–A83 (2021). [3] B. Fornberg, N. Flyer, Solving PDEs with radial basis functions, Acta Numer. 24, 215–258 (2015).