A Multipole Expansion Method for PDE Constrained Problems

Abstract

It is crucial to choose the appropriate numerical method for treating partial differential equations in shape optimization and control problems. This paper introduces a meshless approach derived from the well-known charge simulation method. Instead of a large number of heuristically located monopoles (i.e. charges or sources), the proposed technique relies on more rigorously located poles with multiplicity. A well-conditioned method is devised by applying basis orthogonalization in this multipole expansion. The basis size is determined by a recursive process of orthogonalization in order to achieve the desired accuracy as shown in the numerical examples.