Efficient convolution with the Newton potential in d dimensions

The paper is concerned with the evaluation of the convolution integral R R d 1 x−y f (y)dy in d dimensions (usually d = 3), when f is given as piecewise polynomial of possibly large degree, i.e., f may be considered as an hp-finite element function. The underlying grid is locally refined using various levels of dyadically organised grids. The result of the convolution is approximated in the same kind of mesh. If f is given in tensor product form, the d-dimensional convolution can be reduced to one-dimensional convolutions. Although the details are given for the kernel 1/ x , the basis techniques can be generalised to homogeneous kernels, e.g., the fundamential solution const·x 2−d of the d-dimensional Poisson equation.