Isogeometric preconditioners based on fast solvers for the Sylvester equation

TL;DR

This paper introduces a preconditioning method based on solving Sylvester-like equations that leverages tensor structures in isogeometric discretizations, improving the efficiency of solving large Poisson problems.

Contribution

It proposes a robust preconditioning strategy using Sylvester equation solutions that exploits tensor structures in isogeometric analysis, applicable to high-degree splines and complex geometries.

Findings

The method is robust against mesh size and spline degree variations.

Both direct and iterative solvers for the Sylvester equation are effective.

Numerical experiments demonstrate the approach's potential for various geometries.

Abstract

We consider large linear systems arising from the isogeometric discretization of the Poisson problem on a single-patch domain. The numerical solution of such systems is considered a challenging task, particularly when the degree of the splines employed as basis functions is high. We consider a preconditioning strategy which is based on the solution of a Sylvester-like equation at each step of an iterative solver. We show that this strategy, which fully exploits the tensor structure that underlies isogeometric problems, is robust with respect to both mesh size and spline degree, although it may suffer from the presence of complicated geometry or coefficients. We consider two popular solvers for the Sylvester equation, a direct one and an iterative one, and we discuss in detail their implementation and efficiency for 2D and 3D problems on single-patch or conforming multi-patch NURBS geometries. Numerical experiments for problems with different domain geometries are presented, which demonstrate the potential of this approach.