Dual-Primal Isogeometric Tearing and Interconnecting methods for the Stokes problem

TL;DR

This paper develops and tests dual-primal IETI-DP methods tailored for the saddle point structure of the Stokes problem in isogeometric analysis, addressing challenges like null-space and preconditioning.

Contribution

It extends IETI-DP methods to the Stokes problem, analyzing preconditioners and primal degrees of freedom choices for saddle point systems.

Findings

Different scaled Dirichlet preconditioners tested successfully.

Method applied to simple and complex geometries.

Addressed null-space issues in pressure modes.

Abstract

We are interested in a fast solver for linear systems obtained by discretizing the Stokes problem with multi-patch Isogeometric Analysis. We use Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) methods. In resent years, IETI-DPand related methods have been studied extensively, mainly for the Poisson problem. For the Stokes equations, several challenges arise since the corresponding system is not positive definite, but has saddle point structure. Moreover, the Stokes equations with Dirichlet boundary conditions have a null-space, consisting of the constant pressure modes. This poses a challenge when considering the scaled Dirichlet preconditioner. We test out two different scaled Dirichlet preconditioners with different choices of primal degrees of freedom. The tests are performed on rather simple domains (the unit square and a quarter annulus) and a more complicated domain (a Yeti-footprint).