Speaker: Jingfang Huang (UNC Chapel Hill)
Title: On the Algebraic Structure of Layer and Volume Potentials
In this talk, we study the algebraic structures of the Laplace layer and volume potentials with smooth density function and boundary geometry for an evaluation point w inside a Fast Multipole Method (FMM) leaf box centered at w_c. When the box is well separated from the boundary, the potentials can be approximated by a local polynomial expansion p_n(w-w_c) and the errors follow standard FMM far-field error analysis. When the box is located close to or on the boundary, nonlinear contribution from the boundary can be accurately captured using the zeros of a polynomial equation. Understanding these algebraic structures allows for higher order numerical discretizations of both layer and volume potentials. We present preliminary numerical
results to demonstrate the accuracy of the resulting quadrature by two expansions (QB2X) representations for different potentials.