Domain decomposition methods with small overlap for Qn−Qn−2 spectral elements

An indefinite overlapping Schwarz method is introduced and studied for Stokes problems discretized with Qn−Qn−2 spectral elements. This results in a parallel and scalable preconditioner for the iterative solution of the discrete system of equations. The preconditioner is based on the solution of local Stokes problems on overlapping subregions and a coarse Stokes problem with Q2−Q0 elements. The efficiency of the method is based on using a nodal basis and quadrature rules associated with Gauss–Lobatto–Legendre nodes. As for h-version finite elements, a small overlap between subregions seems to be the most efficient choice.