Domain Decomposition Methods in Science and Engineering

FETI205 A conforming FEM for (2.7) results in the discrete optimality system K1 0 BT1 00 K̃2−B2 −C2 ΥB1 −B2 0 00−ΥC2 0 Υ  u1u2λμ = f1f200(2.8) where K̃2 ≡ K2+C2 ΥC2.Elimination of the primal variables in (2.8) results in the coarse grid problem[B1K−11 B T1 +B2K̃−12 B T2 B2K̃−12 CT2 ΥΥC2K̃−12 B T2ΥC2K̃−12 CT2 Υ−Υ] [λμ]=[d1d1](2.9) where[d1d2]=[B1K−11 f1 −B2K̃−12 f2−ΥC2K̃−12 f2]. As compared with FETI-1, the columns of K̃−12 CT2 Υ are approximating a basis for therigid body modes associated withΩ2, and K̃−12 is an approximation to the pseudoinverse ofK2. Inserting the solution of the coarse grid problem (2.9) into (2.8) results in K1u1 = f1−B1 λ(2.10)K̃2u2 = f2 +B T2 λ+CT2 Υμ.(2.11) These two linear systems have symmetric positive definite coefficient matrices and can besolved in parallel.We remark that (2.11) corresponds to the minimization problem infv∈H1(Ω2)12ã(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 amodification of FETI-1 that allows for a wider choice of well-posed primal problems for thesedomains. In particular, our approach results in nonsingular linear systems with propertiesthat can be easily controlled.Our starting point is the splitting ofH(Ω2) into the direct sumH(Ω2) = H 1c (Ω2)⊕N2 whereN2 is the RBM space for Ω2 andHc (Ω2) = {u ∈H(Ω2) | c2(u) = 0}, is the complement space with respect to the moments c2. The report [1] demonstrates thatsuch 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 forN2 and α ∈ R. To solve (2.1)we consider the problem of finding the saddle-point(u1, u2c, α, λ) ∈H(Ω1, ∂Ω1)×Hc (Ω2)×Rp ×H(Γ) of the Lagrangian L(û1, û2c, α̂, λ̂) =2∑ i=1(12a(ûi, ûi)Ωi − 〈f, ûi〉Ωi)+ 〈λ̂, û1 − (û2c +R2α̂)〉Γ. (3.1) 206BOCHEV, LEHOUCQ This Lagrangian only differs from the FETI-1 Lagrangian by explicitly specifying a particularsolution on the floating subdomain. The optimality system for (3.1) is to seek(u1, u2c, α, λ) ∈H(Ω1, ∂Ω1)×Hc (Ω2)× R ×H(Γ) such thata(û1, u1)Ω1 +〈û1, λ〉Γ = 〈f, û1〉Ω1 ∀û1 ∈H(Ω1, ∂Ω1)a(û2c, u2c)Ω2 − 〈û2c, λ〉Γ = 〈f, û2c〉Ω2 ∀û2c ∈ Hc (Ω2)−〈R2α̂, λ〉Γ=〈f,R2α̂〉Ω2 ∀α̂ ∈ R〈λ̂, u1 − (u2c +R2α)〉Γ = 0 ∀λ̂ ∈ H(Γ).(3.2) Note that in (3.2) the floating subdomain problem is restricted to finding a particular solutionout of the complement space Hc (Ω2) rather than the space H(Ω2). This seemingly minorchange makes the floating subdomain problem uniquely solvable. Therefore, its conformingdiscretization, that is restriction to a finite element subspace of Hc (Ω2), would engendera non-singular linear system. However, building a finite element subspace of Hc (Ω2) maynot be a simple matter and discretization by standard finite element subspaces ofH(Ω2) ispreferred.To enable the use of standard finite elements the floating subdomain equation is furtherreplaced by a regularized problem in which the bilinear form a(·, ·)Ω2 is augmented by the termc2(û2)Υc2(u2). The regularized optimality system is to seek (u1, u2, α, λ) ∈H(Ω1, ∂Ω1)×H(Ω2)× R ×H(Γ) such that a(û1, u1)Ω1 +〈û1, λ〉Γ= 〈f, û1〉Ω1 ∀û1 ∈H(Ω1, ∂Ω1)a(û2, u2)Ω2 + c2(û2)Υc2(u2)− 〈û2, λ〉Γ = 〈f, û2〉Ω2 ∀û2 ∈H(Ω2)−〈R2α̂, λ〉Γ=〈f,R2α̂〉Ω2 ∀α̂ ∈ R〈λ̂, u1 − (u2 +Rα)〉Γ= 0 ∀λ̂ ∈ H(Γ).(3.3) Theorem 3.1 Problems (3.2) and (3.3) are equivalent. Proof. The only point that needs to be verified is that a solution (u1, u2, α, λ) of (3.3) hasits second component in the complement space Hc (Ω2). Choosing û2 = R2α̂ in the secondequation in (3.3) combined with the third equation gives c2(R2α̂)Υc2(u2) =〈R2α̂, λ〉Γ +〈f,R2α̂〉Ω2 ≡ 0for any α̂ ∈ R. Therefore, c2(u2) = 0 and u2 ∈ Hc (Ω2). A conforming FEM for (3.3) results in the linear system K1 0BT100 K̃2−B200 0−(B2R2) 0B1 −B2 0−B2R2  u1u2λα = f1f2R2 f20(3.4) where K̃2 is the same matrix as in (2.8) and we redundantly use R2 to denote the coefficientsassociated with the finite element approximants for the RBMs.We note the close similarity between (3.4) and a FETI-1 discrete problem. In both casesa particular solution for the floating subdomain is generated and a component inN2 is addedto satisfy the interface continuity condition. However, in contrast to a FETI-1, in (3.4) thefloating subdomain matrix is non-singular and we have complete control over the choice ofthe particular solution by virtue of the momentsc2. These moments can be further selectedso as to optimize the nonsingular matrix K̃2 with respect to a particular solver. ABSTRACT-FETI207FETI207 Elimination of the primal variables in (3.4) results in the coarse grid problem[B1K−11 B T1 +B2K̃−12 B T2 −B2R−(B2R)0] [λα]=[d1d1](3.5) where[d1d2]=[B1K−11 f1 −B2K̃−12 f2R f2]. Inserting the solution of the coarse grid problem (3.5) into (3.4) results in K1u1 = f1−B1 λ(3.6)K̃2u2 = f2 +B T2 λ.(3.7) This primal system and the FETI-1 primal system only differ in the coefficient matrix foru2. Here K̃2 is symmetric positive definite whereas FETI-1 uses the singular K2. Thereforea computation of a pseudoinverse is avoided.4. The moments c(·). Suppose that we have a floating subdomain Ω, a RBM sub-space N and resulting basis R (discrete or continuous). The moments c(·) play a centralrole in our regularization strategy. Both of the FETI formulations introduced in this reportrely upon these moments to regularize the floating subdomain problems. The purpose of themoments is to provide an “energy” measure for the RBMs that otherwise have zero strainenergy a(·, ·).Therefore, the guiding principle in their choice is to ensure that they form a non-degenerate set. By non-degenerate here we mean that the matrix c(R) of order p is non-singular. For linear elasticity [1] one such set of moments is given by the functional c(v) ≡ ∫ΩΘ1v∫ ΩΘ2∇× v(4.1) where the diagonal elements of Θ1 = diag(θ1,1, θ1,2, θ1,3) and Θ2 = diag(θ2,1, θ2,2, θ2,3)(4.2) are elements ofH−1(Ω) satisfying the hypothesis∫ Ωθ1,i = 0 and∫