Chebyshev collocation solutions of flow problems

Abstract

Chebyshev collocation methods are reviewed. For general second-order elliptic equations, the algebraic system obtained through the collocation process is ill-conditioned. Recent work on finite element preconditioning shows that bilinear elements give full satisfaction. For Stokes problems, as interpolants of different degree are used for the velocity and the pressure, the classical nine-node Lagrangian element with biquadratic velocities and bilinear pressures constitutes the best choice. In order to treat the non-linear terms, a Newton's method is designed. Domain decomposition is set up with the jumps of the stress vector across interfaces between adjacent subdomains. Two-dimensional curvy geometries are handled by a coordinate mapping. Problems with singularities are treated by a mixed finite element and spectral approximation. Finally, current developments for Non-Newtonian fluids are evoked.