Computation of volume potentials over bounded domains via approximate approximations

TL;DR

This paper introduces a new method for computing volume potentials over bounded domains by combining approximate approximations with tangential-halfspace integration, resulting in highly accurate and efficient numerical formulas.

Contribution

It develops cubature formulas that simplify volume potential computation to one-dimensional quadratures using approximate approximation basis functions.

Findings

Formulas provide high accuracy in numerical tests.

Confirmed the predicted order of convergence.

Efficient reduction to one-dimensional integrals.

Abstract

We obtain cubature formulas of volume potentials over bounded domains combining the basis functions introduced in the theory of approximate approximations with their integration over the tangential-halfspace. Then the computation is reduced to the quadrature of one dimensional integrals over the halfline. We conclude the paper providing numerical tests which show that these formulas give very accurate approximations and confirm the predicted order of convergence.