A direct solver for the least-squares spectral collocation system on rectangular elements for the incompressible Navier-Stokes equations
The authors develop a least-square spectral collocation scheme for steady and unsteady Stokes equations and for incompressible Navier-Stokes equations where the original domain has been decomposed into quadrilateral subelements and on the element interfaces continuity of the functions is enforced. To avoid high condition numbers of normal equations, a direct solver (QR decomposition of the matrices) is used for the overdetermined systems. The numerical simulations confirm the high accuracy of the proposed scheme, and solving the overdetermined systems with QR decomposition yields better approximation results. The computational cost of QR decomposition is higher than using normal equations. At the same time, by using pseudoinverse Clenshaw-Curtis quadrature rule one gets the same results as by using QR decomposition, but with much higher computational costs. For unsteady Stokes equations, the presented scheme also demonstrates a good performance for different Reynolds numbers, for different time steps and for various numbers of elements. Finally, the scheme is successfully applied to the regularized lid-driven cavity problems where good results are achieved.
scientific article; zbMATH DE number 5276048
scientific article