Deflation-based preconditioning for immersed finite element methods and immersogeometric analysis
For researchers and practitioners using immersed finite element methods, this work provides a more robust preconditioning approach to handle ill-conditioned systems caused by small cut elements.
This work identifies limitations of existing preconditioning strategies for immersed finite element methods with small cut elements, and proposes a robust deflation-based preconditioning technique that improves conditioning and solver performance.
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.