Periodic Reporting for period 1 - NEMESIS (NEw generation MEthods for numerical SImulationS)
Reporting period: 2024-01-01 to 2025-06-30
Relevant partial differential equation (PDE) problems of the 21st century, including those encountered in magnetohydrodynamics and geological flows, involve severe difficulties linked to:
* the presence of incomplete differential operators related to Hilbert complexes;
* nonlinear and hybrid-dimensional physical behaviors;
* embedded/moving interfaces.
The goal of the NEMESIS project is to lay the groundwork for a novel generation of numerical simulators tackling all of the above difficulties at once. This requires the combination of skills and knowledge resulting from the synergy of the Principal Investigators, which cover distinct and extremely technical fields of mathematics: numerical analysis, analysis of nonlinear PDEs, and scientific computing.
The research program is structured into four tightly interconnected clusters, whose goals are:
* the development of Polytopal Exterior Calculus (PEC), a general theory of discrete Hilbert complexes on polytopal meshes;
* the design of innovative strategies to boost efficiency, embedded into a general abstract Multilevel Solvers Convergence Framework (MSCF);
* the extension of the above tools to challenging nonlinear and hybrid-dimensional problems through Discrete Functional Analysis (DFA) tools;
* the demonstration through proof-of-concept applications in magnetohydrodynamics (e.g. nuclear reactor models or aluminum smelting) and geological flows (e.g. flows of gas/liquid mixtures in underground reservoirs with fractures, as occurring in CO2 storage).
This project will bring key advances in numerical analysis through the introduction of entirely novel paradigms such as PEC and DFA, and in scientific computing through MSCF. The novel mathematical tools developed in the project will break long-standing barriers in engineering and applied sciences, and will be implemented in a practitioner-oriented open-source library that will boost design and prediction capabilities in these fields.
* the presence of incomplete differential operators related to Hilbert complexes;
* nonlinear and hybrid-dimensional physical behaviors;
* embedded/moving interfaces.
The goal of the NEMESIS project is to lay the groundwork for a novel generation of numerical simulators tackling all of the above difficulties at once. This requires the combination of skills and knowledge resulting from the synergy of the Principal Investigators, which cover distinct and extremely technical fields of mathematics: numerical analysis, analysis of nonlinear PDEs, and scientific computing.
The research program is structured into four tightly interconnected clusters, whose goals are:
* the development of Polytopal Exterior Calculus (PEC), a general theory of discrete Hilbert complexes on polytopal meshes;
* the design of innovative strategies to boost efficiency, embedded into a general abstract Multilevel Solvers Convergence Framework (MSCF);
* the extension of the above tools to challenging nonlinear and hybrid-dimensional problems through Discrete Functional Analysis (DFA) tools;
* the demonstration through proof-of-concept applications in magnetohydrodynamics (e.g. nuclear reactor models or aluminum smelting) and geological flows (e.g. flows of gas/liquid mixtures in underground reservoirs with fractures, as occurring in CO2 storage).
This project will bring key advances in numerical analysis through the introduction of entirely novel paradigms such as PEC and DFA, and in scientific computing through MSCF. The novel mathematical tools developed in the project will break long-standing barriers in engineering and applied sciences, and will be implemented in a practitioner-oriented open-source library that will boost design and prediction capabilities in these fields.
**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(opens in new window)).
**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(opens in new window)) 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/(opens in new window)).
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(opens in new window)).
**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(opens in new window)) 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/(opens in new window)).
Polytopal Exterior Calculus (PEC) is a major result beyond the state of the art. It is indeed the first time that fully discrete and computable de Rham complexes of differential forms were developed and analyzed. PEC will serve as a stepping stone for future developments, including, in particular, the design of discrete advanced complexes on general polytopal meshes.
The polytopal trace theory is a genuinely innovative result, for which we had to design a suitable discrete H^{1/2}-like boundary norm that was strong enough to yield a lifting but weak enough to allow for trace inequalities. Beyond the application to BDDC solvers, we expect it to enable complete analyses of many domain-decomposition methods for polytopal schemes.
Finite and Virtual Element/HHO methods for magnetohydrodynamics (MHD) and related models that are pressure robust and convection robust (both with respect to the fluid and magnetic Reynolds numbers) have not been tackled in the literature. The various advancements in this first stage of the project (robust elements for linearized MHD, advanced time-discretization techniques, local Péclet numbers analysis) set up the stage for a major result beyond the state of the art in the simulation of the fully nonlinear MHD equations.
Finally, significant progress has been made in developing and analysing new numerical methods for complex multiphysics phenomena and integrating machine learning paradigms to enhance accuracy and efficiency. We have devised innovative hybrid simulation approaches that provide great flexibility in handling domains with irregular and heterogeneous geometries and ensure physics robustness. The methodology has been validated, focusing on key models that arise in both geophysical and life sciences, capturing complex interactions and biomechanical behaviour.
The polytopal trace theory is a genuinely innovative result, for which we had to design a suitable discrete H^{1/2}-like boundary norm that was strong enough to yield a lifting but weak enough to allow for trace inequalities. Beyond the application to BDDC solvers, we expect it to enable complete analyses of many domain-decomposition methods for polytopal schemes.
Finite and Virtual Element/HHO methods for magnetohydrodynamics (MHD) and related models that are pressure robust and convection robust (both with respect to the fluid and magnetic Reynolds numbers) have not been tackled in the literature. The various advancements in this first stage of the project (robust elements for linearized MHD, advanced time-discretization techniques, local Péclet numbers analysis) set up the stage for a major result beyond the state of the art in the simulation of the fully nonlinear MHD equations.
Finally, significant progress has been made in developing and analysing new numerical methods for complex multiphysics phenomena and integrating machine learning paradigms to enhance accuracy and efficiency. We have devised innovative hybrid simulation approaches that provide great flexibility in handling domains with irregular and heterogeneous geometries and ensure physics robustness. The methodology has been validated, focusing on key models that arise in both geophysical and life sciences, capturing complex interactions and biomechanical behaviour.