A Cartesian non-uniform grid interpolation method for fast field evaluation on elongated domains

An algorithm for performing a fast evaluation on time harmonic fields radiated by two-dimensional source constellations with elongated geometry is introduced. The suggested scheme attains a linear computational complexity. At the core of the algorithm is the idea of phase and amplitude compensation facilitating the conversion of fields radiated by spatially confined sources into bandlimited functions of the suitable coordinates. Thus, the fields or potentials are represented by their samples on sparse left-side and right-side non-uniform Cartesian grids. These grids are employed both for the representation of outgoing fields and as targets for accumulating the incoming fields. Integrating this approach with the divide-and-conquer strategy, the algorithm relies on a binary hierarchical decomposition of the computational domain. The total field is computed via a two-way multilevel process, which passes through all of the left-side and right-side grids, while systematically aggregating all subdomains' contributions, ending with the field interpolation back to the desired observation points.