Multilevel particle-partition of unity method

This paper is concerned with the particle-partition of unity method, a meshfree generalization of the finite element method. We present the fundamental construction principles and abstract approximation properties of the resulting function spaces V ^PU. Moreover, we discuss the construction of optimal approximation spaces for a reference application in linear elastic fracture mechanics in particular. The presented construction not only yields optimal convergence rates globally independently of the regularity of the solution, our method shows a super-convergence near the singular points of the solution.