A Multilevel Schwarz Preconditioner Based on a Hierarchy of Robust Coarse Spaces

A parallel implementation of a multilevel linear solver, part of the HPDDM library and tested in the FreeFEM language.

The code available in this repository can reproduce the results from the following paper.

@article{aldaas2019multi,
    Author = {Al Daas, Hussam and Grigori, Laura and Jolivet, Pierre and Tournier, Pierre-Henri},
    Title = {A Multilevel {Schwarz} Preconditioner Based on a Hierarchy of Robust Coarse Spaces},
    Year = {2021},
    Journal = {SIAM Journal on Scientific Computing},
    Volume = {43},
    Number = {3},
    Pages = {A1907--A1928},
    Url = {https://github.com/prj-/aldaas2019multi}
}

Getting started

Dependencies

Make sure you have access to a recent FreeFEM installation (version 4.3 or above), compiled with the hpddm plugin. More details about the HPDDM options used in the solver may be found in the cheat sheet.

Usage example

One should be able to launch the following commands, which solves the scalar diffusion or linear elasticity equations on the same geometrical configurations as in the paper.

$ mpirun -np 8 FreeFem++-mpi diffusion-2d.edp -v 0
$ mpirun -np 8 FreeFem++-mpi diffusion-3d.edp -v 0
$ mpirun -np 8 FreeFem++-mpi elasticity-2d.edp -v 0
$ mpirun -np 8 FreeFem++-mpi elasticity-3d.edp -v 0

The option -v 0 is here to minimize the output generated by FreeFEM, see this tutorial for more information. Some timings are stored in a subfolder as .txt files, and it is possible to postprocess solutions using ParaView.

Using the previous commands, one should get four convergence histories per .edp script. The first (resp. second) one corresponds to the linear system being solved with an exact LU decomposition of the level 2 coarse operator with two (resp. four) MPI processes. The third (resp. fourth) convergence history corresponds to the linear system being solved with the multilevel approach proposed in the paper using two (resp. four) superdomains.

Customization

Here are the main command line parameters common to all scripts.

• -global (default to different values depending on the script), number of discretization points of the global domain
• -split (default to 1), number of local uniform refinement steps after the global domain has been decomposed
• -base (default to 2), sets how the number of superdomains varies: baseⁱ, for i = 1, ..., 4
• -no_timing, if used, nothing is written in the subdirectory timing/
• -output, if used, a .pvd and various .vtu are generated in the subdirectory output/
• -Dpetsc=true, if used, the systems are solved using PETSc instead of HPDDM. In particular, options are provided for PCHYPRE and PCGAMG

Acknowledgements

• European Commission Horizon 2020 programme, project NLAFET, grant number 671633
• HPC resources of TGCC@CEA under the allocation A0050607519 made by GENCI