Abstract: We will describe two separate but related methodologies that have been implemented in the open source, finite element code ASPECT, which computational geophysicists use to model a wide variety of problems that arise in Earth and Planetary geophysics. The first technique is a volume-of-fluid (VOF) interface tracking algorithm that was originally designed to model the subduction of the oceanic lithosphere of a tectonic plate beneath a less dense lithosphere of a second plate. However, we have since used this VOF methodology to model some basic laboratory experiments in order to benchmark some of the rheological models that have been implemented in ASPECT. The second technique is the particle or particle-in-cell (PIC) methodology, which we have been developing and benchmarking for use in ASPECT for the past seven years or so. This PIC methodology has been shown to have excellent weak and strong scaling over at least three orders of magnitude of model size on a uniform grid. In addition, our PIC algorithm shows that strong scaling for the adaptive grid case is nearly as good as for the uniform grid case, decreasing the total runtime essentially linearly from 96 to 3,072 cores. We will briefly show a collection of benchmarks we have used and developed to assess the accuracy of this PIC methodology and conclude with a description and video of a beam bending in a less dense viscoelastic medium due to the force of gravity in which the viscoelastic rheology is modeled by the components of stress that are carried on the particles and interpolated onto the underlying finite element grid at each time step.