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 compute incompressible viscous flows by means of finite element approximations. In the proposed algorithm, the convection, the diffusion, and the incompressibility are treated in three different substeps. The convection is treated first by means of a Lagrange--Galerkin technique, whereas the diffusion and the incompressibility are treated separately in two subsequent substeps by means of a projection method. It is shown that provided the time step, δ t, is of O(hd/4), where h is the meshsize and d is the space dimension (2≤ d ≤ 3), the proposed method yields for finite time T an error of O(hˡ⁺¹+δ t) in the L2 norm for the velocity and an error of O(hˡ+δ t) in the H1 norm (or the L2 norm for the pressure), where l is the polynomial degree of the approximate velocity.