Efficient Direct Methods for Solving the Spectral Collocation Equations for Stokes Flow in Rectangularly Decomposable Domains
Timothy N. Phillips, Andréas Karageorghis · SIAM Journal on Scientific and Statistical Computing · 1989
A spectral element method is described for solving Stokes flow in rectangularly decomposable domains. The flow region is divided into a number of rectangular subregions, some of which may be semi-infinite. The solution to the governing biharmonic equation for the stream function is represented by an expansion of Chebyshev polynomials in each subregion. The coefficients in these expansions are determined by collocating the differential equation and boundary conditions, and imposing $C^3 $ continuity across subregion interfaces. The spectral element method produces systems of equations that are block tridiagonal. Efficient direct methods based on capacitance matrix ideas are proposed that take advantage of the structure of the spectral element matrix. The use of these techniques produces savings in storage and time.