Isogeometric preconditioners based on fast solvers for the Sylvester equation
This work addresses the challenge of solving large linear systems from isogeometric analysis with high-degree splines, offering a robust preconditioner for practitioners in computational mechanics.
The paper proposes a preconditioning strategy for isogeometric discretizations of the Poisson problem that solves a Sylvester-like equation at each iterative step, showing robustness to mesh size and spline degree. Numerical experiments demonstrate its potential for 2D and 3D problems on single-patch and multi-patch NURBS geometries.
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.