We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
38/2007
Convolution of $hp$-functions on locally refined grids - extended version
Wolfgang Hackbusch
Abstract
Usually, the fast evaluation of a convolution integral $\int_{\mathbb{R}}f(y)g(x-y)\mathrm{d}y$ requires that the functions $f,g$ have a simple structure based on an equidistant grid in order to apply the fast Fourier transform. Here we discuss the efficient performance of the convolution of $hp$-functions in certain locally refined grids. More precisely, the convolution result is projected into some given $hp$-space (Galerkin approximation). The overall cost is $\mathcal{O}(p^{2}N\log N),$ where $N$ is the sum of the dimensions of the subspaces containing $f$, $g$ and the resulting function, while $p$ is the maximal polynomial degree.
