Convergence of the grid-free point vortex method for the three-dimensional Euler equations

Convergence of the grid-free point vortex method is proved for three-dimensional Euler equations with smooth solutions. Two new techniques are used to obtain consistency and stability of the method. The first one is Strang's trick which allows smooth approximate solutions to be constructed to the vortex method equations with arbitrarily small errors. This result is used to obtain consistency and nonlinear stability. The second tool is the use of a very special discrete negative norm in l₁ space for vorticity which gives rise to the linear stability result. Combining these two techniques proves uniform convergence of the method with second-order accuracy.