A Mixed-Basis Spectral Projection Method

A new grid-less (no collocation) spectral projection method is presented. The unsteady Navier–Stokes equations are approximated according to the variational framework of Guermond and Quartapelle which accommodates two vector spaces for the velocity fields obtained in the two half-steps of the fractional-step method but retains only one in the final solution algorithm. Two different bases built on Legendre polynomials are used for the velocity and pressure to solve the corresponding Helmholtz and Poisson equations by direct spectral elliptic solvers. Interpolations PN and PN−2 are employed for velocity and pressure to satisfy the LBB stability requirement and a Gauss–Legendre quadrature formula with 32N integration points is used to prevent aliasing error in the pseudospectral evaluation of the nonlinear terms. A BDF second-order time stepping is implemented to provide accurate numerical results about the stability of the singular driven cavity problem.