Regularized formulations of FETI

FETI 205 A conforming FEM for (2.7) results in the discrete optimality system   K1 0 B T 1 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 ] . As compared with FETI-1, the columns of K̃−1 2 C T 2 Υ are approximating a basis for the rigid body modes associated with Ω2, and K̃ −1 2 is an approximation to the pseudoinverse of K2. Inserting the solution of the coarse grid problem (2.9) into (2.8) results in K1u1 = f1 −B1 λ (2.10) K̃2u2 = f2 +B T 2 λ+C T 2 Υμ. (2.11) These two linear systems have symmetric positive definite coefficient matrices and can be solved in parallel. We remark that (2.11) corresponds to the minimization problem inf v∈H1(Ω2) 1 2 ã(v, v)− 〈f̃ , v〉Ω2 where f̃ is the continuous load associated with the discrete load of (2.11). 3. FETI-SS: Regularization by space splitting. In this section we introduce a modification of FETI-1 that allows for a wider choice of well-posed primal problems for these domains. In particular, our approach results in nonsingular linear systems with properties that can be easily controlled. Our starting point is the splitting of H(Ω2) into the direct sum H(Ω2) = H 1 c (Ω2)⊕N2 where N2 is the RBM space for Ω2 and H c (Ω2) = {u ∈ H(Ω2) | c2(u) = 0}, is the complement space with respect to the moments c2. The report [1] demonstrates that such a splitting exists for any non-degenerate set of moments. As a result, any u2 ∈ H(Ω2) can be uniquely written as u2c +R2α where R2 is a basis for N2 and α ∈ R. To solve (2.1) we consider the problem of finding the saddle-point (u1, u2c, α, λ) ∈ H(Ω1, ∂Ω1)×H c (Ω2)× R p ×H(Γ) of the Lagrangian L(û1, û2c, α̂, λ̂) = 2 ∑ i=1 (1 2 a(ûi, ûi)Ωi − 〈f, ûi〉Ωi ) + 〈λ̂, û1 − (û2c +R2α̂)〉Γ. (3.1)