Preconditioning immersed isogeometric finite element methods with application to flow problems

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.

Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

“…Trimmed basis functions on small cut elements -which can be almost linearly dependent -are also assigned to blocks associated to other functions. This satisfies the requirement formulated in [50] that almost linearly dependent basis functions need to be in a block together, and therefore resolves the small eigenmodes that are characteristic for immersed Figure 8a displays the support within the physical domain of the function associated to the block in orange. Figure 8b displays the supports of the other functions assigned to the block in purple.…”

“…This is consistent with the analysis of the conditioning problems in [35], which points out that diagonal preconditioners do not adequately mitigate the almost linear dependencies that occur in immersed finite element methods. In [50] it is derived that almost linearly dependent basis functions can be effectively treated collectively when these are inverted in a block manner by a Schwarz-type method. Based on the additive Schwarz lemma, [50] shows that additive Schwarz preconditioning is actually a very natural approach to resolve the small eigenmodes caused by almost linear dependencies.…”

“…References [50] and [51] establish that additive Schwarz preconditioners can effectively resolve the conditioning problems of immersed finite element methods with higher-order discretizations, for both isogeometric and hp-finite element function spaces and for both symmetric positive definite (SPD) and non-SPD problems. Furthermore, the numerical investigation in [50] conveys that the conditioning of immersed systems treated by an additive Schwarz preconditioner is very similar to that of mesh-fitting systems. In particular, the condition number of an additive Schwarz preconditioned immersed system exhibits the same mesh-size dependence as mesh-fitting approaches [52,53], which opens the doors to the application of established concepts of multigrid preconditioning.…”

“…The intrinsic dependence on mesh regularity in geometric multigrid approaches is non-restrictive for immersed finite element methods, as these methods generally employ structured grids. Based on the observations regarding the mesh-size dependence of the additive Schwarz preconditioner developed in [50], this contribution further investigates the spectral properties of immersed systems treated with Schwarz-type preconditioners, in order to establish the suitability of these as smoothers in a multigrid method. This results in a preconditioning technique with the desired properties, the performance of which is demonstrated on a range of test cases with multi-million degrees of freedom.…”

“…This contribution only considers SPD problems. Multigrid methods are an established concept in fluid mechanics as well [60], however, and similar Schwarz-type methods have successfully been applied to flow problems with both immersed finite element methods [50] and mesh-fitting multigrid solvers [71], i.e., Vanka-smoothers [79]. Therefore, it is anticipated that the presented preconditioning technique extends matatis mutandis to non-SPD and mixed formulations.…”