# Scalable multigrid methods for immersed finite element methods and immersed isogeometric analysis

## TL;DR

This paper develops a scalable multigrid preconditioner for immersed finite element and isogeometric analysis methods, effectively addressing ill-conditioning issues and enabling efficient large-scale parallel computations.

## Contribution

It introduces a geometric multigrid preconditioner that is mesh- and cut-element-independent, suitable for higher-order discretizations and various basis functions.

## Key findings

- Provides mesh-independent convergence rates.
- Enables linear-scaling parallel solutions.
- Effective for higher-order and locally refined discretizations.

## Abstract

Ill-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral properties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems in parallel, at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines.