The vortex method with finite elements

This work shows that the method of charcteristics is well suited for the numerical solution of first order hyperbolic partial differential equations whose coefficients are approximated by functions piecewise constant on a finite element triangulation of the domain of integration. We apply this method to the numerical solution of Euler’s equation and prove convergence when the time step and the mesh size tend to zero. The proof is based upon the results of regularity given by Kato and Wolibner and on L ∞ {L^\infty } estimates for the solution of the Dirichlet problem given by Nitsche. The method obtained belongs to the family of vortex methods usually studied in a finite difference context.