**Polytopal complexes**
In the first 18 months, we laid the foundation of Polytopal Exterior Calculus by generalizing Discrete de Rham and Virtual Element complexes via differential forms. This enabled unified proofs of key results like discrete Poincaré inequalities and adjoint consistency, previously derived through operator-specific arguments.
We also initiated a systematic extension of the fully discrete approach to extended complexes. Notably, we applied the Bernstein-Gelfand-Gelfand construction to polytopal complexes for the first time, deriving a two-dimensional strain complex.
These discrete complexes were implemented in the HArDCore library (
https://github.com/jdroniou/HArDCore(odnośnik otworzy się w nowym oknie)).
**Efficiency boost**
We developed novel hybrid approaches combining physics-based models and machine learning to enhance large-scale simulations through:
i) mesh agglomeration;
ii) improved convergence of multilevel algebraic solvers;
iii) automated artificial viscosity models for conservation laws.
The newly released MAGNET library (
https://github.com/lymphlib/magnet(odnośnik otworzy się w nowym oknie)) uses Graph Neural Networks for mesh agglomeration, currently explored for efficient multigrid preconditioners and adaptive domain partitions.
**Taming physical complexity**
A major theoretical achievement was the first complete trace theory for polytopal methods, yielding both trace inequalities and a lifting property. This was used in the first analysis of BDDC linear solvers for polytopal schemes.
For interface problems, we developed a method for diffusion with jump interface conditions, paving the way for moving interface models. We also derived a posteriori error estimates for a contact model using flux equilibration.
We began developing an exterior-calculus version of the cut-finite element method, suitable for non-body-fitted meshes, laying the foundation for cut-Polytopal Exterior Calculus, particularly relevant for curved domains.
In the context of hybrid-dimensional problems, we designed a higher-order variant of our bubble-polytopal method for elasticity with contact conditions (e.g. in fractured porous media). Theoretical analysis shows improved accuracy and robustness over the lowest-order formulation.
**Proof-of-concept applications**
We made significant progress in the development and analysis of robust numerical schemes for advanced fluid dynamics.
We studied pre-asymptotic convergence based on local dimensionless numbers for various incompressible flow models, deriving pressure- and Reynolds-semi-robust estimates for schemes for the full Navier–Stokes equations. Particular emphasis was placed on convection-dominated flows, starting from scalar problems up to resistive MHD equations.
Advanced time-discretization techniques were introduced to efficiently handle non-stationary fluid dynamics. Non-Newtonian Stokes flows were also addressed, with tailored stabilization and accurate polyhedral schemes. A key focus was on designing methods that preserve conserved quantities in MHD, like helicity and cross-helicity. Much of this work was integrated into the VEM++ library.
Another central effort was the design, analysis, and testing of high-order polytopal methods for nonlinear multiphysics and multiscale PDEs. These schemes were developed to ensure physical robustness, structural fidelity, and geometric adaptability. Applications include geophysical and biomedical models, such as thermo-hydro-mechanical coupling in soil exploitation, non-Newtonian flows in manufacturing, and fluid dynamics of biological tissues.
These developments were incorporated into the Lymph library (
https://lymph.bitbucket.io/(odnośnik otworzy się w nowym oknie)).
