A weak Legendre collocation spectral method for the solution of the incompressible Navier–Stokes equations in unstructured quadrilateral subdomains

Abstract

A weak Legendre spectral method is developed for the solution of the primitive variable formulation of the unsteady incompressible Navier–Stokes equations, in general two-dimensional and axisymmetric geometries. A semi-implicit projection method is utilized for the temporal approximation and for decoupling the velocity field from the pressure field. A series of elliptic boundary value problems arises from the above procedure, each of which is spatially discretized by a weak collocation method in multiple nonoverlapping subdomains. In particular, a modified variational formulation of the partial differential equations is presented which leads, after discretization, to a weak multidomain approximation of the corresponding problems. A weak formalism for the influence matrix technique is also developed, which is consistent with the spatial discretization scheme and successfully separate the equations for the internal nodes from the ones governing the interface unknowns. A method of dealing with the singularity problem faced by the weak formulation at axisymmetric problems is proposed, while a combination of direct methods is studied for tackling effectively the linear algebraic systems resulting from the full discretization. Exponential convergence is demonstrated for a plethora of Stokes and Navier–Stokes simulations.