Multipole expansion solution of the Laplace equation using surface data

This paper provides a computational method to model a three-dimensional static electromagnetic field within a finite source free volume starting from discrete field information on its surface. The method uses the Helmholtz vector decomposition theorem and the differential algebraic framework of COSY INFINITY to determine a solution to the Laplace equation. The solution is locally expressed as a Taylor expansion of the field which can be computed to arbitrary order. It provides a natural multipole decomposition of the field which is required for the computation of transfer maps, and also allows to obtain very accurate finite element representations with very small numbers of cells.