Fast solvers for robust discretisations in CFD – Fast4HHO

The project "Fast solvers for robust discretisations in CFD" will provide scalable, robust linear solvers for Compatible Discrete Operator (CDO) and Hybrid High Order (HHO) discretisations in industrial computational fluid dynamics (CFD) applications.
The CDO and HHO discretisations combine the following, desirable properties:
-) They obtain optimal convergence orders even on distorted meshes.
-) They apply to general, polyhedral meshes.
Additionally, in the case of the HHO methods:
-) They provide a formalism to construct approximations of arbitrary order.
-) They allow a static condensation step that reduces the size of the system matrix, which makes the approach computationally appealing compared to other hybrid discontinuous Galerkin (HDG) schemes.

These discretisation schemes are in the process of being introduced into the open source software Code_Saturne, which is developed at EDF for its CFD purposes. The numerical simulation of fine scale fluid phenomena on complex geometries with Code_Saturne requires regularly meshes of the order of tens of millions, up to hundreds of millions of cells. In order to accommodate the complex geometries, the meshes are unstructured and, in general, composed of arbitrary polyhedra. In an effort to keep the computational requirements in terms of computing power and run times acceptable, the robust CDO and HHO discretisations need to be paired with fast, scalable, equally robust and algebraic solvers.

This project brings together expertise in
-) HHO discretisations (IMAG) and CDO discretisations (EDF),
-) linear algebra for saddle-point problems (IRIT, CERFACS),
-) multigrid methods for very large Stokes problems (CERFACS),
-) construction of robust aggregation multigrid methods in CFD applications (EDF), and
-) software development practices for industrial use (EDF)
in order to derive algebraic multigrid methods (AMG) for CDO and HHO applications in industrial CFD.


The project will treat the question of suitable fast solvers for the differential operators in CFD in the following order:
1. scalar diffusion,
2. convection-diffusion,
3. Stokes
in an effort to provide the building blocks for the extension to the Navier-Stokes system. For each operator, we will start with lowest-order approximations before moving up in polynomial order.

In the course of the project, we will investigate the relevance of recent results in the academic literature concerning
a) the transformation of the discretised Stokes operator, obtained by continuous finite elements, which
allows the efficient resolution with an aggregation AMG with standard smoothers, and
b) the construction and performance of p-multigrid methods for HDG schemes.

All competitive solutions that will be developed in the course of this project will be implemented and made available in the open source software Code_Saturne.