We’re excited to share two recent preprints on arXiv authored by our consortium partners, showcasing advances in solving large-scale problems in scientific computing. These contributions push the boundaries of efficiency and scalability in finite element simulations – a core focus of the dealii-X project.
In “Optimal and Scalable Augmented Lagrangian Preconditioners for Fictitious Domain Problems“, the authors tackle a classic computational challenge: solving large, structured systems of equations arising from immersed boundary methods and fictitious domain techniques.
Their solution? Two novel augmented Lagrangian preconditioners, designed to work seamlessly with flexible GMRES solvers. These preconditioners significantly boost convergence rates and are backed by spectral analysis for both exact and approximate forms. Numerical experiments – including Poisson and Stokes problems in 2D and 3D – demonstrate both robustness and scalability of their approach.
The second preprint, “Matrix-Free Implementation of the Non-Nested Multigrid Method“, reimagines multigrid for complex meshes. While traditional multigrid methods rely on nested mesh hierarchies, this work introduces a non-nested, matrix-free, and parallel implementation within the C++ finite-element library deal.II.
By decoupling the hierarchy, this method provides more flexibility—especially for highly refined or unstructured meshes—and still delivers strong performance across a range of problems, including Poisson and linear elasticity in 2D and 3D. It supports multiple polynomial degrees and complex geometries.
Both papers underline our shared mission: building scalable, flexible, and high-performance tools for modern scientific computing. We’re proud to see these innovations emerge from within the dealii-X community! Stay tuned for deeper dives into the technical details, and don’t forget to check out the full papers on arXiv:
https://arxiv.org/abs/2412.10910
https://arxiv.org/abs/2504.11339