Asynchronous variational integration (AVI) is a tool which improves the numerical efficiency of explicit time stepping schemes when applied to finite element meshes with local spatial refinement. This is achieved by associating an individual time step length to each spatial domain. Furthermore, long-term stability is ensured by its variational structure. This article presents AVI in the context of finite elements based on a weakened weak form (W2) Liu (2009) , exemplified by continuous assumed gradient elements Wolff and Bucher (2011) . The article presents the main ideas of the modified AVI, gives implementation notes and a recipe for estimating the critical time step.