Symbolic preconditioning techniques for linear systems of partial differential equations

Thomas Cluzeau (1) , Victorita Dolean (2) , Frédéric Nataf (3) , Alban Quadrat (4, 5)

1 XLIM-DMI - DMI
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 LJLL - Laboratoire Jacques-Louis Lions
4 DISCO - Dynamical Interconnected Systems in COmplex Environments
5 L2S - Laboratoire des signaux et systèmes

Some algorithmic aspects of systems of PDEs based simulations can be better clarified by means of symbolic computation techniques. This is very important since numerical simulations heavily rely on solving systems of PDEs. For the large-scale problems we deal with in today's standard applications, it is necessary to rely on iterative Krylov methods that are scalable (i.e., weakly dependent on the number of degrees on freedom and number of subdomains) and have limited memory requirements. They are preconditioned by domain decomposition methods, incomplete factorizations and multigrid preconditioners. These techniques are well understood and efficient for scalar symmetric equations (e.g., Laplacian, biLaplacian) and to some extent for non-symmetric equations (e.g., convection-diffusion). But they have poor performances and lack robustness when used for symmetric systems of PDEs, and even more so for non-symmetric complex systems (fluid mechanics, porous media ...). As a general rule, the study of iterative solvers for systems of PDEs as opposed to scalar PDEs is an underdeveloped subject. We aim at building new robust and efficient solvers, such as domain decomposition methods and preconditioners for some linear and well-known systems of PDEs.

Domaines: Analyse numérique [math.NA]

Thomas Cluzeau, Victorita Dolean, Frédéric Nataf, Alban Quadrat. Symbolic preconditioning techniques for linear systems of partial differential equations. 2012. ⟨hal-00664092v3⟩