Preconditioning immersed isogeometric finite element methods with application to flow problems

TL;DR

This paper extends preconditioning techniques for immersed isogeometric finite element methods to non-SPD and mixed flow problems, introducing a connectivity-based additive-Schwarz preconditioner that improves numerical stability and efficiency.

Contribution

It generalizes the CbAS preconditioner to non-SPD and mixed problems, enhancing the applicability of immersed finite element methods for complex flow simulations.

Findings

CbAS preconditioner effectively improves solver convergence

Enhanced stability for non-SPD and mixed flow problems

Numerical results demonstrate improved computational efficiency

Abstract

Immersed finite element methods generally suffer from conditioning problems when cut elements intersect the physical domain only on a small fraction of their volume. De Prenter et al. [Computer Methods in Applied Mechanics and Engineering, 316 (2017) pp. 297-327] present an analysis for symmetric positive definite (SPD) immersed problems, and for this class of problems an algebraic preconditioner is developed. In this contribution the conditioning analysis is extended to immersed finite element methods for systems that are not SPD and the preconditioning technique is generalized to a connectivity-based preconditioner inspired by Additive-Schwarz preconditioning. This Connectivity-based Additive-Schwarz (CbAS) preconditioner is applicable to problems that are not SPD and to mixed problems, such as the Stokes and Navier-Stokes equations. A detailed numerical investigation of the effectivity of the CbAS preconditioner to a range of flow problems is presented.