Spectral analysis of the stiffness matrix sequence in the approximated Stokes equation

TL;DR

This paper investigates the spectral properties of matrices from Taylor-Hood discretizations of variable viscosity Stokes problems, providing theoretical insights, numerical tests, and implications for preconditioning.

Contribution

It offers a detailed spectral analysis of the stiffness matrix sequence in variable viscosity Stokes equations, including localization, distributional results, and preconditioning considerations.

Findings

Spectral features depend on variable viscosity properties.

Numerical tests confirm theoretical spectral distribution results.

Preconditioning strategies can be informed by spectral analysis.

Abstract

In the present paper, we analyze in detail the spectral features of the matrix sequences arising from the Taylor-Hood $\mathbb{P}_2$-$\mathbb{P}_1$ approximation of variable viscosity for $2d$ Stokes problem under weak assumptions on the regularity of the diffusion. Localization and distributional spectral results are provided, accompanied by numerical tests and visualizations. A preliminary study of the impact of our findings on the preconditioning problem is also presented. A final section with concluding remarks and open problems ends the current work.