A Multipole Expansion Method for PDE Constrained Problems

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.