A direct solver for the least-squares spectral collocation system on rectangular elements for the incompressible Navier–Stokes equations

A least-squares spectral collocation scheme for the Stokes and incompressible Navier–Stokes equations is proposed. The original domain is decomposed into quadrilateral subelements and on the element interfaces continuity of the functions is enforced in the least-squares sense. The collocation conditions and the interface conditions lead to overdetermined systems. These systems are directly solved by QR decomposition of the underlying matrices. By numerical simulations it is shown that the direct method leads to better results than the approach with normal equations. Furthermore, it is shown that the condition numbers can be reduced by introducing the Clenshaw–Curtis quadrature rule for imposing the average pressure to be zero. Finally, our scheme is successfully applied to the regularized and lid-driven cavity flow problems.