Augmented Lagrangian preconditioners for fictitious domain formulations of elliptic interface problems

TL;DR

This paper introduces a new augmented Lagrangian preconditioner for elliptic interface problems with jump coefficients, enhancing convergence and computational efficiency in finite element solutions.

Contribution

It develops a novel AL preconditioner based on Fictitious Domain methods, including a cheaper variant, and provides theoretical and numerical validation of its effectiveness.

Findings

Eigenvalue clustering for the ideal preconditioner

Mesh-independent iteration counts

Robustness over large coefficient jumps

Abstract

We present a novel augmented Lagrangian (AL) preconditioner for the solution of linear systems arising from finite element discretizations of elliptic interface problems with jump coefficients. The method is based on the Fictitious Domain with Distributed Lagrange Multipliers formulation and it is designed to improve the convergence of the Flexible Generalized Minimal Residual (FGMRES) method in the presence of large coefficient jumps. To reduce the computational cost, we also introduce a cheaper block-triangular variant of the preconditioner. We prove eigenvalue clustering for the ideal AL preconditioner and study the limiting behavior of the spectrum for the modified variant in terms of parameters and the size of the jumps. Numerical experiments on different immersed geometries confirm mesh-independent iteration counts and robustness over large coefficient jumps, with substantial reductions in wall-clock time for the modified approach.