Convergence analysis of a finite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations

This paper provides a convergence analysis of a fractional-step method to c ompute incompressible viscous flows by means of finite element approximatio ns. In the proposed algorithm, the convection, the diffusion, and the incom pressibility are treated in three different substeps. The convection is tre ated first by means of a Lagrange-Galerkin technique, whereas the diffusion and the incompressibility are treated separately in two subsequent substep s by means of a projection method. It is shown that provided the time step, delta t, is of O(h(d/4)), where h is the meshsize and d is the space dimen sion (2 less than or equal to d less than or equal to 3), the proposed meth od yields for finite time T an error of O(h(l+1) + delta t) in the L-2 norm for the velocity and an error of O(h(l) + delta t) in the H-1 norm (or the L-2 norm for the pressure), where l is the polynomial degree of the approx imate velocity.