Robust isogeometric preconditioners for the Stokes system based on the Fast Diagonalization method

TL;DR

This paper introduces a new class of robust preconditioners for the isogeometric discretization of the Stokes system, utilizing the Fast Diagonalization method to efficiently solve Sylvester-like equations, improving solver speed on complex domains.

Contribution

The paper presents a novel preconditioning approach for the Stokes system that remains effective across different spline degrees and mesh sizes, incorporating geometry and coefficient information for efficiency.

Findings

Preconditioners are robust with respect to spline degree and mesh size.

The proposed method significantly accelerates iterative solver convergence.

Efficiency is maintained on complex geometries and variable viscosity domains.

Abstract

In this paper we propose a new class of preconditioners for the isogeometric discretization of the Stokes system. Their application involves the solution of a Sylvester-like equation, which can be done efficiently thanks to the Fast Diagonalization method. These preconditioners are robust with respect to both the spline degree and mesh size. By incorporating information on the geometry parametrization and equation coefficients, we maintain efficiency on non-trivial computational domains and for variable kinematic viscosity. In our numerical tests we compare to a standard approach, showing that the overall iterative solver based on our preconditioners is significantly faster.