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Recently, a mixed finite element-least square point interpolation method (FE-LSPIM) has been extended to deal with 2D acoustic problem by the authors. That element employed radial-polynomial basis functions for the local approximation (LA). This paper presents a FE-Meshfree QUAD4 element for analyzing the two dimensions (2D) and three dimensions (3D) acoustic problem by combining the excellent property of FE and radial-polynomial basis point interpolation shape functions by utilizing the partition of unity (PU) principles methods. In this work, the acoustic domain is discretized by quadrilateral mesh, and then the shape functions of quadrilateral element and the radial point interpolation are used for the LA. The radial-polynomial basis capacitates the proposed method to free from the possible singularity of the moment matrix that could sometimes result with an inappropriate choice of polynomial basis functions. The present method also offers an “appropriate-stiff” model for reducing the numerical dispersion error as compared to the previous FEM or FE-LSPIM with pure polynomial basis, especially for the high wave number problem. These findings have been validated through several numerical test problems. [ABSTRACT FROM AUTHOR]