In this paper, we present a parallel multigrid PDE solver working on adaptive hierarchical cartesian grids. The presentation is restricted to the linear elliptic operator of second order, but extensions are possible and have already been realised as prototypes. Within the solver the handling of the vertices and the degrees of freedom associated to them is implemented solely using stacks and iterates of a Peano space–filling curve. Thus, due to the structuredness of the grid, two administrative bits per vertex are sufficient to store both geometry and grid refinement information. The implementation and parallel extension, using a space–filling curve to obtain a load balanced domain decomposition, will be formalised. In view of the fact that we are using a multigrid solver of linear complexity \documentclass12pt{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{O}(n)$\end{document}, it has to be ensured that communication cost and, hence, the parallel algorithm’s overall complexity do not exceed this linear behaviour.