Boundary layer approximate approximations and cubature of potentials in domains

In this article we present a new approach to the computation of volume potentials over bounded domains, which is based on the quasi-interpolation of the density by smooth, almost locally supported basis functions for which the corresponding volume potentials are known. The quasi-interpolant is a linear combination of the basis function with shifted and scaled arguments and with coe cients explicitly given by the point values of the density. Thus, the approach results in semi-analytic cubature formulae for volume potentials, which prove to be high order approximations of the integrals. It is based on multi-resolution schemes for accurate approximations up to the boundary by applying approximate re nement equations of the basis functions and iterative approximations on ner grids. We obtain asymptotic error estimates for the quasi-interpolation and corresponding cubature formulae and provide some numerical examples.