A matrix-theoretic spectral analysis of incompressible Navier-Stokes staggered DG approximation and related solvers

TL;DR

This paper conducts a detailed spectral analysis of the linear systems arising from a staggered DG discretization of incompressible Navier-Stokes equations, aiming to inform the design of efficient iterative solvers.

Contribution

It introduces a matrix-theoretic spectral analysis leveraging Toeplitz and block structures to improve solver design for Navier-Stokes discretizations.

Findings

Spectral properties of the matrices are characterized using Toeplitz and tensor structures.

The analysis informs the development of faster iterative solvers.

Numerical experiments demonstrate promising solver performance on complex geometries.

Abstract

The incompressible Navier-Stokes equations are solved in a channel, using a Discontinuous Galerkin method over staggered grids. The resulting linear systems are studied both in terms of the structure and in terms of the spectral features of the related coefficient matrices. In fact, the resulting matrices are of block type, each block showing Toeplitz-like, band, and tensor structure at the same time. Using this rich matrix-theoretic information and the Toeplitz, Generalized Locally Toeplitz technology, a quite complete spectral analysis is presented, with the target of designing and analyzing fast iterative solvers for the associated large linear systems. Quite promising numerical results are presented, commented, and critically discussed for elongated two- and three-dimensional geometries.