A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations The numerical treatment of partial differential equations with meshfree discretization techniques has been a very active research area in recent years. Up to now however meshfree methods have been in an early experimental stage and were not competitive due to the lack of efficient iterative solvers and numerical quadrature. This volume now presents an efficient parallel implementation of a meshfree method namely the partition of unity method (PUM). A general numerical integration scheme is presented for the efficient assembly of the stiffness matrix as well as an optimal multilevel solver for the arising linear system. Furthermore detailed information on the parallel implementation of the method on distributed memory computers is provided and numerical results are presented in two and three space dimensions with linear higher order and augmented approximation spaces with up to 42 million degrees of freedom