







NIMS Workshop and Minicourse :Mathematical Analysis, Numerics and Application in fluid and gas dynamics' 








Title 


Speaker 
Hyea Hyun Kim 


Date 
20090203 


Host 



Place 
KAIST 




Abstract : A scalable FETIDP (DualPrimal Finite Element Tearing and Interconnecting) algorithm for the Stokes problem is developed and analyzed. FETIDP methods are known to be the most scalable domain decomposition algorithm which has been tested up to 1000 processors and successfully applied to fast solutions of engineering problems with 10 million degrees of freedom. There have been considerable attempts for development of scalable domain decomposition algorithms for the Stokes problem. The limitation of all the previous approaches is the use of primal unknowns which are selected from both the velocity unknowns and the pressure unknowns [6, 3, 4, 5]. The primal unknowns in the domain decomposition algorithm are related to the coarse problem which globally corrects residual errors to achieve a good scalability in terms of the number of processors, i.e., the number of subdomains. A certain infsup stability is required for the selection of the primal unknowns in the previous approaches with both the velocity and the pressure primal unknowns. This usually results in a relatively large number of primal unknowns and quite complicated implementation of the algorithms in three dimensional problems. In our new approach, we introduce the primal unknowns selected from the velocity unknowns. The advantages of the new approach are a more stable coarse problem without primal pressure unknowns and the use of relatively cheap lumped preconditioner. In two dimensions, the velocity unknowns at subdomain corners are selected as the primal unknowns and the condition number bound, CH/h(1 + log(H/h)), is proved, see [1]. Especially in three dimensions, these advantages provide more robust and faster FETIDP algorithm. In three dimensions, the velocity unknowns at subdomain corners and the averages of velocity on common faces are selected as the primal unknowns in the FETIDP formulation. The condition number bound is analyzed to be CH/h, where C is independent of any mesh parameters and H/h is the number of elements across a subdomain, see [2]. 





