MiS Preprint Repository

Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.

MiS Preprint

Efficient convolution with the Newton potential in $d$ dimensions

Wolfgang Hackbusch


The paper is concerned with the evaluation of the convolution integral $\int_{\mathbb{R}^{d}}\frac{1}{\left\Vert x-y\right\Vert }f(y)\mathrm{d}y$ 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 $1/\left\Vert x\right\Vert ,$ the basis techniques can be generalised to homogeneous kernels, e.g., the fundamential solution $const\cdot\left\Vert x\right\Vert ^{2-d}$ of the $d$-dimensional Poisson equation.

Sep 4, 2007
Sep 4, 2007
newton potential, convolution, coulomb potential

Related publications

2008 Repository Open Access
Wolfgang Hackbusch

Efficient convolution with the Newton potential in d dimensions

In: Numerische Mathematik, 110 (2008) 4, pp. 449-489