Adaptive Mesh Refinement for Singular Solutions of the Incompressible Euler Equations

The occurrence of a finite time singularity in the incompressible Euler equations in three dimensions is studied numerically using the technique of adaptive mesh refinement. As opposed to earlier treatments, a prescribed accuracy is guaranteed over the entire integration domain. A singularity in the vorticity could be traced down to five levels of refinement which corresponds to a resolution of ${2048}^3$ mesh points in a nonadaptive treatment. The growth of vorticity fits a power law behavior proportional to $1/(T^{*}-t)$ where $T^{*}$ denotes the time when the singularity occurs.