Stable discretizations and IETI-DP solvers for the Stokes system in multi-patch Isogeometric Analysis

TL;DR

This paper develops a stable discretization and an efficient IETI-DP solver for the Stokes system in multi-patch Isogeometric Analysis, ensuring robustness across complex geometries.

Contribution

It extends stability results of the Taylor--Hood discretization from single-patch to multi-patch domains and introduces a geometry-robust IETI-DP solver.

Findings

The stability of the discretization depends on geometry shape.

The proposed IETI-DP solver is robust and converges reliably.

Numerical tests confirm theoretical convergence and stability.

Abstract

We are interested in a fast solver for the Stokes equations, discretized with multi-patch Isogeometric Analysis. In the last years, several inf-sup stable discretizations for the Stokes problem have been proposed, often the analysis was restricted to single-patch domains. We focus on one of the simplest approaches, the isogeometric Taylor--Hood element. We show how stability results for single-patch domains can be carried over to multi-patch domains. While this is possible, the stability strongly depends on the shape of the geometry. We construct a Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solver that does not suffer from that effect. We give a convergence analysis and provide numerical tests.