High-order numerical evaluation of volume potentials via polynomial density interpolation

This short note outlines a simple numerical method for the high-order numerical evaluation of volume integral operators associated with the Poisson and Helmholtz equations in two spatial dimensions. Following the ideas of the density interpolation method for boundary integral operators, the proposed methodology leverages Green's third identity and a local polynomial interpolation of the source function to recast the volume potential as a sum of single- and double-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated everywhere in the plane by means of existing methods (e.g. the density interpolation method), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules. Preliminary numerical examples based on non-overlapping quadrilateral patch representations of the domain in conjunction with Chebyshev-grid discretizations demonstrate the effectiveness of the proposed approach. Direct extensions of this method to other polynomial interpolation strategies in two and three dimensions and the efficient implementation via fast algorithms, will be presented in the complete version of this preliminary paper.