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The goal of this paper is to create a fruitful bridge between the numericalmethods for approximating partial differential equations (PDEs) in fluiddynamics and the (iterative) numerical methods for dealing with the resultinglarge linear systems. Among the main objectives are the design of new efficientiterative solvers and a rigorous analysis of their convergence speed. The linkwe have in mind is either the structure or the hidden structure that theinvolved coefficient matrices inherit, both from the continuous PDE and fromthe approximation scheme: in turn, the resulting structure is used for deducingspectral information, crucial for the conditioning and convergence analysis,and for the design of more efficient solvers.  As specific problem we consider the incompressible Navier-Stokes equations,as numerical technique we consider a novel family of high order accurateDiscontinuous Galerkin methods on staggered meshes, and as tools we use thetheory of Toeplitz matrices generated by a function (in the most general block,multi-level form) and the more recent theory of Generalized Locally Toeplitzmatrix-sequences. We arrive at a quite complete picture of the spectralfeatures of the underlying matrices and this information is employed for givinga forecast of the convergence history of the conjugate gradient method,together with a discussion on new more advanced techniques (involvingpreconditioning, multigrid, multi-iterative solvers). Several numerical testsare provided and critically illustrated in order to show the validity and thepotential of our analysis.

Staggered discontinuous Galerkin methods for the incompressible Navier-Stokes equations: spectral analysis and computational results