Localized Hermite Method of Approximate Particular Solutions for Solving the Helmholtz Equation

This paper proposes a localized Hermite method of approximate particular solutions (LHMAPS) for solving the 2D inhomogeneous Helmholtz-type equations. Building on the local scheme of the localized method of approximate particular solutions (LMAPS) for the Helmholtz-type differential operator, LHMAPS employs Hermite-type local approximations involving both the solution values and their Laplacian to improve the accuracy of LMAPS. The polyharmonic spline (PS) radial basis functions and polynomial basis functions are considered in the formulation of LHMAPS. Numerical experiments are presented to demonstrate the enhanced accuracy achieved by employing Hermite-type local approximations.