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Section: New Results

Resolution of linearised systems

Participants : Olivier Allain [Lemma] , Gautier Brèthes, Alain Dervieux, Bruno Koobus, Stephen Wornom.

For Fluid Mechanics as well as for Structural Mechanics, implicit time-advancing is mandatory. It can be applied efficiently if the large systems involved are solved with a good parallel algorithm. In the 90's, a generation of solution algorithms was devised on the basis of Domain Decomposition Methods (DDM). For complex models such as compressible flows, Schwarz DDM were combined with quasi-Newton algorithms like GMRES. For example in the Aironum tool, we use Restrictive Additive Schwarz (RAS, developed by Cai and Farhat). RAS is an ancestor of the widely used class of Newton-Krylov-Schwarz (NKS) algorithms. NKS are, in some versions including RAS, almost scalable i.e. their convergence rate is independant of the number of processors. But scalability degrades over a thousand processors. During the ANR ECINADS, coordinated by Ecuador, a Coarse-Grid Deflated RAS was developed. The algorithmic scalability (iteration-wise) holds for all part, except for the coarse grid direct solver, which concerns a much smaller problem. Effective Convergence Scalability (ECS) was confirmed up to 2048 processors. After this level, the asymptotic complexity of the coarse-grid direct solver become predominant and ECS is lost. In other words, with a Coarse-Grid Deflated RAS, the size of the coarse grid problem must be limited in order to enjoy ECS.

In the thesis of Gautier Brèthes, we now study a further step towards super-massive scalability: we use a number of fine and medium grids in order to solve the target large system by a multi-mesh multigrid (MG) algorithm. An important novelty is that the complete FMG technology is applied, with a new stopping criterion avoiding useless cycling [12] . It is well-known that parallel MG is limited to “no-too-coarse” coarse levels due to an excessive ratio between communication and computation. Now our parallel MG can be complemented by the previous version of the solver (deflated RAS) for this no-too-coarse coarse level.