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MiS Preprint
81/2007
Efficient convolution with the Newton potential in $d$ dimensions
Wolfgang Hackbusch
Abstract
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.
