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

TL;DR

This paper develops spectral analysis and efficient iterative solvers for high-order staggered discontinuous Galerkin methods applied to the incompressible Navier-Stokes equations, linking PDE discretization with linear system solution strategies.

Contribution

It introduces a novel spectral analysis framework for the matrices from staggered DG discretizations and proposes optimized iterative solvers based on this analysis.

Findings

Spectral properties of the matrices are fully characterized.

Conjugate gradient method convergence can be predicted from spectral data.

Numerical tests demonstrate the effectiveness of the proposed methods.

Abstract

The goal of this paper is to create a fruitful bridge between the numerical methods for approximating partial differential equations (PDEs) in fluid dynamics and the (iterative) numerical methods for dealing with the resulting large linear systems. Among the main objectives are the design of new efficient iterative solvers and a rigorous analysis of their convergence speed. The link we have in mind is either the structure or the hidden structure that the involved coefficient matrices inherit, both from the continuous PDE and from the approximation scheme: in turn, the resulting structure is used for deducing spectral 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 accurate Discontinuous Galerkin methods on staggered meshes, and as tools we use the theory of Toeplitz matrices generated by a function (in the most general block, multi-level form) and the more recent theory of Generalized Locally Toeplitz matrix-sequences. We arrive at a quite complete picture of the spectral features of the underlying matrices and this information is employed for giving a forecast of the convergence history of the conjugate gradient method, together with a discussion on new more advanced techniques (involving preconditioning, multigrid, multi-iterative solvers). Several numerical tests are provided and critically illustrated in order to show the validity and the potential of our analysis.