A Preconditioner for the Feti-dp Formulation of the Stokes Problem with Mortar Methods

We consider a FETI-DP formulation for the Stokes problem on nonmatching grids in 2D. The FETI-DP method is a domain decomposition method that uses Lagrange multipliers to match the solutions continuously across the subdomain boundaries in the sense of dual-primal variables. We use the mortar matching condition as the continuity constraints for the FETI-DP formulation. Moreover, to satisfy the compatibility condition of the local Stokes problem and to solve the Stokes problem efficiently, redundant continuity constraints are introduced. Lagrange multipliers corresponding to the redundant constraints are treated as primal variables in the FETI-DP formulation. We propose a preconditioner for the FETI-DP operator, which is derived from a dual norm on the Lagrange multiplier space. The dual norm is obtained from a duality pairing between the Lagrange multiplier space and the velocity function space restricted on the nonmortar sides. Then, we show that the condition number of the preconditioned FETI-DP operator is bounded by C maxi=1,··· ,N { (1 + log (Hi/hi)) 2 } , where Hi and hi are the subdomain size and the mesh size, respectively, and C is a constant independent of Hi’s and hi’s.