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We investigate the formation and early evolution and fragmentation of an accretion disk around a forming massive protostar. We use a grid-based self-gravity-radiation-hydrodynamics code including a sub-grid module for stellar and dust evolution. On purpose, we do not use sink particles to allow for all paths of fragment formation and destruction, but instead keeping the spatial grid resolution high enough to properly resolve the physical length scales of the problem. We use a 3D grid in spherical coordinates with a logarithmic scaling in the radial direction and cosine scaling in the polar direction. Because of that, roughly 25% of the total number of grid cells, corresponding to $\sim$ 26 million grid cells, are used to model the disk physics. They constitute the highest resolution simulations performed up to now on disk fragmentation around a forming massive star with the physics considered here. We study the convergence of our results by performing the same simulation for 5 different resolutions. We start from the collapse of a molecular cloud; a massive (proto)star is formed in its center, surrounded by a fragmenting Keplerian-like accretion disk with spiral arms. The fragments have masses of $\sim 1 M_\odot$, and their continuous interactions with the disk, spiral arms and other fragments results in eccentric orbits. Fragments form hydrostatic cores, surrounded by secondary disks with spiral arms that also produce new fragments. We identified several mechanisms of fragment formation, interaction and destruction. Central temperatures of the fragments can reach the hydrogen dissociation limit, form second Larson cores and evolve into companion stars. Based on this, we study the multiplicity predicted by the simulations and find $\sim 6$ companions at different distances from the primary: from possible spectroscopic multiples, to companions at distances between 1000 and 2000 au.