1 . Regularized formulations of FETI

FETI 3 Theorem 2.1 (u1, u2, λ) solves (2.4) if and only if (u1, u2, λ, μ = c(u2)) solves (2.7). Proof. The theorem is easily established by using the stabilized constraints (2.5) and recalling that τ = 0. The theorem demonstrates that (2.5) represents a consistent stabilization. The impact of this innocuous sleight of hand is that the resulting coarse grid problem is equivalently stabilized. We now demonstrate this. A conforming FEM for (2.7) results in the discrete optimality system   K1 0 B1 0 0 K̃2 −B2 −C2 Υ B1 −B2 0 0 0 −ΥC2 0 Υ     u1 u2 λ μ   =   f1 f2 0 0   (2.8) where K̃2 ≡ K2 + C2 ΥC2. Elimination of the primal variables in (2.8) results in the coarse grid problem [ B1K−1 1 B T 1 + B2K̃ −1 2 B T 2 B2K̃ −1 2 C T 2 Υ ΥC2K̃−1 2 B T 2 ΥC2K̃ −1 2 C T 2 Υ−Υ ] [ λ μ ] = [ d1 d1 ] (2.9) where [ d1 d2 ] = [ B1K−1 1 f1 −B2K̃−1 2 f2 −ΥC2K̃−1 2 f2 ]