Chebyshev collocation solutions of the navier-stokes equations using multi-domain decomposition and finite element preconditioning

Abstract

The steady Navier-Stokes equations are solved using series of basis functions involving Chebyshev polynomials. The projection method is a collocation scheme. A Newton's linearization is performed in order to obtain a set of algebraic equations. As the matrix system is ill conditioned, the collocation technique is preconditioned by a standard Galerkin finite element method using a 9-nodes Lagrangian element which presents decisive advantages: sparsity, reduced condition number, easy treatment of complicated geometries. To handle nontrivial geometries in the collocation process, a domain decomposition is set up. The treatment of interface conditions is fully described. Several test problems like the regularized driven square cavity and the backward facing step are discussed to show the abilities of the present algorithm.