  Solvers for partial differential equations (PDE) are one of the cornerstones
of computational science. For large problems, they involve huge amounts of data
that needs to be stored and transmitted on all levels of the memory hierarchy.
Often, bandwidth is the limiting factor due to relatively small arithmetic
intensity, and increasingly so due to the growing disparity between computing
power and bandwidth. Consequently, data compression techniques have been
investigated and tailored towards the specific requirements of PDE solvers
during the last decades. This paper surveys data compression challenges and
corresponding solution approaches for PDE problems, covering all levels of the
memory hierarchy from mass storage up to main memory. Exemplarily, we
illustrate concepts at particular methods, and give references to alternatives.
