A matrix-theoretic spectral analysis of incompressible Navier–Stokes staggered DG approximations and a related spectrally based preconditioning approach

The incompressible Navier–Stokes equations are solved in a channel, using a Discontinuous Galerkin method over staggered grids. We study the structure and the spectral features of the matrices of the linear systems arising from the discretization. They are of block type, each block showing Toeplitz-like, band, and tensor structure at the same time. After introducing new tools to study Toeplitz-like matrix sequences with rectangular symbols, a quite complete spectral analysis is presented, with the target of designing and analyzing fast iterative solvers for the associated large linear systems. Promising numerical results are presented, commented, and critically discussed for elongated two- and three-dimensional geometries.