  A direct numerical simulation of the three-dimensional elektrokinetic
instability near a charge selective surface (electric membrane, electrode, or
system of micro-/nanochannels) is carried out and analyzed. A special
finite-difference method was used for the space discretization along with a
semi-implicit $3\frac{1}{3}$-step Runge-Kutta scheme for the integration in
time. The calculations employed parallel computing. Three characteristic
patterns, which correspond to the overlimiting currents, are observed: (a)
two-dimensional electroconvective rolls, (b) three-dimensional regular
hexagonal structures, and (c) three-dimensional structures of spatiotemporal
chaos, which are a combination of unsteady hexagons, quadrangles and triangles.
The transition from (b) to (c) is accompanied by the generation of interacting
two-dimensional solitary pulses.
