On the selection of primal unknowns for a FETI-DP formulation of the ...

A FETI-DP FORMULATION FOR THE STOKES PROBLEM 5where S is given by⎛⎞K II K I∆ BIT(2.7) S = ⎜⎝KI∆ T K ∆∆ B∆T ⎟⎠ .B I B ∆ 0Substituting (u I , u ∆ , p) into (2.5) and then solving for û Π⎛ ⎞K IΠ(2.8) S ΠΠ û Π = f Π − ⎜⎝K ∆Π⎟⎠B Πwe obtain the resulting algebraic system on λ,(2.9) F DP λ = d,where(2.10)⎛ ⎞0F DP = ⎜⎝J∆T ⎟⎠0⎛ ⎞0d = ⎜⎝J∆T ⎟⎠0TTT⎛⎛⎞ ⎛ ⎞ ⎞f I 0S −1 ⎜⎜⎝⎝f ∆⎟⎠ − ⎜⎝J∆T ⎟⎠ λ ⎟⎠ ,0 0⎛ ⎞ ⎛ ⎞T⎛ ⎞ ⎛ ⎞T⎛ ⎞0 0 K S −1 ⎜⎝J∆T ⎟⎠ + IΠ K IΠ0⎜⎝J∆T ⎟⎠ S −1 ⎜⎝K ∆Π⎟⎠ S−1 ⎜ΠΠ ⎝K ∆Π⎟⎠ S −1 ⎜⎝J T ⎟∆⎠ ,0 0B Π B Π 0⎛⎛⎞ ⎛ ⎞ ⎛ ⎛ ⎞T⎛ ⎞⎞⎞f I K S −1 ⎜⎜⎝⎝f ∆⎟⎠ − IΠK ⎜⎝K ∆Π⎟⎠ S−1 ⎜ΠΠ ⎝ f IΠ f IΠ − ⎜⎝K ∆Π⎟⎠ S −1 ⎜⎝f ∆⎟⎟⎟⎠⎠⎠ ,0 B Π B Π 0and⎛ ⎞K IΠS ΠΠ = K ΠΠ − ⎜⎝K ∆Π⎟⎠B ΠT⎛ ⎞K IΠS −1 ⎜⎝K ∆Π⎟⎠ .B ΠThe resulting system on λ is symmetric and positive semidefinite. In a more detail, whenthe velocity unknowns at subdomain vertices are selected as the primal unknowns, it has onenull space component which is given by(2.11)( )(1) µ ∣∣Γij0µ (2) =0( )ζij n (1)ijζ ij n (2) , ∀Γ ij .ijHere, µ (1)0 and µ (2)0 are Lagrange multipliers related to each x and y- components of velocityunknowns, n (k)ij are each component of n ij , the unit normal vector to Γ ij , and∫(2.12) ζ ij (x l ) = φ l (x(s), y(s)) ds,Γ ij