Augmented Lagrangian preconditioners for fictitious domain formulations of elliptic interface problems
This work addresses computational efficiency for elliptic interface problems in scientific computing, representing an incremental improvement with a cheaper variant.
The authors tackled the problem of solving elliptic interface problems with large coefficient jumps by developing a novel augmented Lagrangian preconditioner for fictitious domain formulations, which achieved mesh-independent iteration counts and substantial reductions in wall-clock time in numerical experiments.
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.