Deflation-based preconditioning for immersed finite element methods and immersogeometric analysis

TL;DR

This paper introduces a robust deflation-based preconditioning technique designed to improve the numerical stability of immersed finite element methods affected by small cut elements.

Contribution

It identifies limitations of existing preconditioners and proposes a new deflation-based approach tailored for immersed finite element methods.

Findings

Existing preconditioners often fail with small cut elements.

The proposed deflation-based preconditioning improves system matrix conditioning.

Counter-examples demonstrate the limitations of traditional strategies.

Abstract

Trimming is a ubiquitous operation in computer-aided-design whereby parts of a geometry are merged, intersected, or simply discarded. While it grants virtually unlimited flexibility in geometric design, it introduces a plethora of other difficulties when such geometries are used within immersed finite element methods. In particular, small cut elements lead to severely ill-conditioned system matrices requiring dedicated penalization, stabilization, or preconditioning techniques. In this work, we highlight the limitations of existing preconditioning strategies by first carefully examining the condition number of the diagonally scaled matrix and later providing realistic counter-examples for some well-established preconditioning strategies. Building on those insights, we propose a robust deflation-based preconditioning technique tailored to immersed finite element methods.