Many engineering problems are described by the fourth-order biharmonic equation (e.g., plate bending problems, Stokes flow etc). These equations also appear in completely different areas, e.g. in scattered data interpolation problems. The traditional solution techniques of these problems are of grid-based (finite difference methods) or mesh-based (finite element methods). In contrast to them, meshfree methods require neither domain nor boundary grid/mesh structures. Such a technique is the Method of Fundamental Solutions [1], which approximates the solution by a linear combination of the fundamental solution of the original equation shifted to some predefined external source points. The coefficients of this linear combination are computed by enforcing the boundary conditions at some predefined boundary collocation points. The method is easily programmable, and its accuracy is often considerably high. However, the resulting linear system of equations is severely ill-conditioned in general, which may cause numerical difficulties. To avoid these computational difficulties, localization techniques have been introduced [2], which convert the original problem into several, smaller subproblems. In this paper, a special localization technique is proposed for the biharmonic equation. The method is similar to the technique proposed in [3]. Based on the classical Schwarz alternating method, the local subproblems are solved separately, resulting in a Seidel-like iterative method. A convergence analysis is detailed for the case in which Navier boundary conditions are assumed. Otherwise, the actual boundary value problem is transformed into an equivalent problem with Navier boundary conditions by solving an auxiliary boundary-only problem. The speed of convergence is significantly increased further by embedding the method into a natural multilevel framework. In addition, the difficulties associated with large, dense, and ill-conditioned linear systems are also avoided. REFERENCES [1] M.A. Golberg, The method of fundamental solutions for Poisson’s equation. Eng. Anal. with Bound. Elements, 16, pp. 205-213, 1995. [2] C.M. Fan, Y.K. Huang, C.S. Chen and S.R. Kuo, Localized method of fundamental solutions for solving two-dimensional Laplace and biharmonic equations. Eng. Anal. with Bound. Elements, 101, pp. 188-97, 2019. [3] C. Gáspár, Biharmonic Scattered Data Interpolation Based on the Method of Fundamental Solutions. LNCS 10476, pp. 485-499. Springer, Cham (2023).