Our new preprints on Parameter robust preconditioning for multi-compartmental Darcy equations and Parameter robust preconditioning by congruence for multiple-network poroelasticity are out on arXiv!
In these studies, we consider systems of equations describing flow in multiple, interacting fluid networks with different permeabilities. The resulting discrete systems of equations can easily become very large, and efficient solution techniques are key. In cases where the inter-network interactions dominate the intra-network permeabilities, we observed that the systems of equations became increasingly hard to solve. In particular, the number of solver iterations would increase with increasing discrete resolution and interaction parameters. However, by creating an algorithm for defining tailored changes of variables, we were able to define robust preconditioners with uniformly bounded iteration counts.
Personal favorite: The lovely trick for how to define parameter-tailored changes of variables via classical matrix analysis including simultaneous diagonalization by congruence.