A unified variational framework for augmented Lagrangian preconditioning of saddle point systems: applications to mixed-dimensional and fictitious domain methods
Abstract
We present a general framework to derive augmented Lagrangian preconditioners for saddle point systems arising from mixed finite element discretizations. Our approach is based on augmenting the variational formulation at the continuous level prior to discretization, providing a unified strategy for designing efficient preconditioners that are robust under mesh refinement for a broad class of problems leading to symmetric saddle point systems. The robustness of the proposed preconditioners is established through a detailed spectral analysis. This framework is applied to different model problems, focusing on mixed-dimensional and fictitious domain formulations involving Lagrange multipliers, with some configurations featuring large coefficient jumps. Extensive numerical experiments on different test cases and geometries are presented, validating the effectiveness and robustness of our preconditioners when used in conjunction with flexible GMRES.
Related
- Accelerating an ensemble of variational data assimilations with randomized preconditioning
- Preconditioning proximal Galerkin: mesh and degree robust solvers for variational inequalities
- Biharmonic Lagrangian surfaces with constant Gaussian curvature in $2$-dimensional complex space forms
- Minimal Lagrangian surfaces in the two-dimensional complex hyperbolic quadric via the loop group method
- A priori bounds for geodesic diameter. Part III. A Sobolev–Poincaré inequality and applications to a variety of geometric variational problems
BibTeX
@misc{benzi2026unified,
title = {{A unified variational framework for augmented Lagrangian preconditioning of saddle point systems: applications to mixed-dimensional and fictitious domain methods}},
author = {Michele Benzi and Marco Feder and Luca Heltai and Federica Mugnaioni},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30357},
eprint = {2610.30357},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30357},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A unified variational framework for augmented Lagrangian preconditioning of saddle point systems: applications to mixed-dimensional and fictitious domain methods
AU - Michele Benzi
AU - Marco Feder
AU - Luca Heltai
AU - Federica Mugnaioni
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30357
UR - https://arxiv.org/abs/2610.30357
AB - We present a general framework to derive augmented Lagrangian preconditioners for saddle point systems arising from mixed finite element discretizations. Our approach is based on augmenting the variational formulation at the continuous level prior to discretization, providing a unified strategy for designing efficient preconditioners that are robust under mesh refinement for a broad class of problems leading to symmetric saddle point systems. The robustness of the proposed preconditioners is established through a detailed spectral analysis. This framework is applied to different model problems, focusing on mixed-dimensional and fictitious domain formulations involving Lagrange multipliers, with some configurations featuring large coefficient jumps. Extensive numerical experiments on different test cases and geometries are presented, validating the effectiveness and robustness of our preconditioners when used in conjunction with flexible GMRES.
ER -