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

• Published in: Numerische Mathematik Vol. 149; no. 4; pp. 933 - 971
• Main Authors: Mazza, M., Semplice, M., Serra-Capizzano, S., Travaglia, E.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2021; Springer Nature B.V
• Subjects: Fluid flow; Galerkin method; Iterative methods; Linear systems; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Matrices (mathematics); Navier-Stokes equations; Numerical Analysis; Numerical and Computational Physics; Preconditioning; Simulation; Spectra; Spectrum analysis; Tensors; Theoretical; 15B05; 65F08; 65N30; 15A18
• ISSN: 0029-599X; 0945-3245; 0945-3245
• DOI: 10.1007/s00211-021-01247-y

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.