Ionized hydrogels, as osmoelastic media, swell enormously (1000 times its original volume in unionized water) due to the osmotic pressure difference caused by the presence of the negatively charged ion groups attached to the solid matrix (polymer chains). The coupling between the extremely large deformations (induced by swelling) and fluid permeation is a field of application that regular poroelasticity formulations cannot handle. In this work, we present a mixed hybrid finite element (MHFE) computational framework featuring a three-field (deformation-chemical potential-flux) formulation. This formulation guarantees that mass conservation is preserved both locally and globally. The impact of such a property on the swelling simulations is demonstrated by four numerical examples in 2D. This paper focuses on the implementation aspects of the MHFE model and shows that it stays robust and accurate for a volume increase of more than 3000%.