An adaptive least-squares spectral collocation method with triangular elements for the incompressible Navier–Stokes equations

A least-squares spectral collocation scheme for the incompressible Navier–Stokes equations is proposed. Grid refinement is performed by means of adaptive triangular elements. On each triangle the Fekete nodes are employed for the collocation of the differential equation. On the element interfaces continuity of the functions is enforced in the least-squares sense. Equal-order Dubiner polynomials are used to obtain a stable spectral scheme. The collocation conditions and the interface conditions lead to an overdetermined system that can be solved efficiently by least-squares. The solution technique only involves symmetric positive-definite linear systems. The approach is first applied to the Poisson equation and then extended to singular perturbation problems where least-squares have a stabilizing effect. By adaptivity, a suitable decomposition of the domain is found where the boundary layer is well resolved. Finally, the method is successfully applied to the regularized driven-cavity flow problem. Numerical simulations confirm the high accuracy of the proposed spectral least-squares scheme.