Least-squares spectral method for velocity-vorticity-pressure form of the Stokes equations

Abstract

The aim of this article is to present and analyze first-order system least-squares spectral method for the Stokes equations in two-dimensional spaces. The Stokes equations are transformed into a first-order system of equations by introducing vorticity as a new variable. The least-squares functional is then defined by summing up the L-w(2)- and H-w(-1) -norms of the residual equations. The H-w(-1)-norm in the least-squares functional is replaced by suitable operator. Continuous and discrete homogeneous least-squares functionals are shown to be equivalent to H-w(1)-norm of velocity and L-w(2)-norm of vorticity and pressure for spectral Galerkin and pseudospectral method. The spectral convergence of the proposed methods are given and the theory is validated by numerical experiment. Mass conservation is also briefly investigated.