Adaptive numerical methods for fluid mechanics

The equations of fluid mechanics can be used to describe natural processes over a wide range of scales, from the behaviour of micro-organisms to astrophysics. Each of these processes is in turn often controlled by internal interactions on widely different scales.  Numerical methods able to efficiently resolve these interactions are — in combination with theoretical analysis and lab experiments — an essential tool for advancing our understanding. I will give a general overview of the hierarchical numerical methods I have worked on, as implemented within the free software Gerris Flow Solver (http://gfs.sf.net) and Basilisk (http://basilisk.fr) and discuss a range of applications including microscale high-energy droplet dynamics, multiphase and complex flows and tsunamis.