Spectral influence matrix for vorticity without corner pathology

Abstract

This paper describes a spectral method for the uncoupled solution of the vorticity and stream function equations in which two-dimensional integral conditions for the vorticity are imposed according to the Glowinski–Pironneau method. A nonsingular influence matrix is obtained which allows to determine the trace of vorticity under no-slip conditions along the entire boundary of a rectangular domain. The method is based on a Galerkin formulation using Legendre polynomials in both directions and fully exploits the direct product structure of the two equations of the Stokes time-discretized problem. Numerical tests for the driven cavity problem are presented to demonstrate that no theoretical or numerical difficulty arises in the proposed 2D spectral approximation, which does not require any a priori regularization of the boundary condition at the corners. The algorithm is found to guarantee convergence to the correct solution for steady and unsteady problems, although spectral accuracy cannot be achieved due to the expected Gibbs' phenomenon.