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We present a family of spacetree-based multigrid realizations using thetree's multiscale nature to derive coarse grids. They align with matrix-freegeometric multigrid solvers as they never assemble the system matrices which iscumbersome for dynamically adaptive grids and full multigrid. The mostsophisticated realizations use BoxMG to construct operator-dependentprolongation and restriction in combination with Galerkin/Petrov-Galerkincoarse-grid operators. This yields robust solvers for nontrivial ellipticproblems. We embed the algebraic, problem- and grid-dependent multigridoperators as stencils into the grid and evaluate all matrix-vector productsin-situ throughout the grid traversals. While such an approach is not literallymatrix-free---the grid carries the matrix---we propose to switch to ahierarchical representation of all operators. Only differences of algebraicoperators to their geometric counterparts are held. These hierarchicaldifferences can be stored and exchanged with small memory footprint. Ourrealizations support arbitrary dynamically adaptive grids while they verticallyintegrate the multilevel operations through spacetree linearization. Thisyields good memory access characteristics, while standard colouring of meshentities with domain decomposition allows us to use parallel manycore clusters.All realization ingredients are detailed such that they can be used by othercodes.

Quasi-matrix-free hybrid multigrid on dynamically adaptive Cartesian grids