ANM for stationary Navier-Stokes equations and with Petrov-Galerkin formulation

Abstract

This paper deals with the use of the asymptotic numerical method (ANM) for solving non-linear problems, with particular emphasis on the stationary Navier–Stokes equation and the Petrov–Galerkin formulation. ANM is a combination of a perturbation technique and a finite element method allowing to transform a non-linear problem into a succession of linear ones that admit the same tangent matrix. This method has been applied with success in non-linear elasticity and fluid mechanics. In this paper, we apply the same kind of technique for solving Navier–Stokes equation with the so-called Petrov–Galerkin weighting. The main difficulty comes from the fact that the non-linearity is no more quadratic and it is not evident, in this case, to be able to compute a large number of terms of the perturbation series. Several examples of fluid mechanic are presented to demonstrate the performance of such a method. Copyright © 2001 John Wiley & Sons, Ltd.