A variational spectral method for the two-dimensional Stokes problem

A spectral Galerkin scheme for solving the two-dimensional stationary Stokes problem is considered. In vorticity-stream function formulation, the Stokes problem is reduced to a biharmonic one. A new formulation for the decomposition of the biharmonic operator is introduced. Main difficulty in this decomposition is to find the trace of the vorticity. Using the adjoint of the composition of trace operator and Greenʹs operator, a variational form is obtained for computing the boundary value of the vorticity. The discrete variational form preserves the properties such as coercivity and continuity of the continuous case which are important for our convergence analysis. Convergence of our scheme is proved, and numerical experiments are performed for the steady driven cavity problem.