An Algebraic Schwarz Theory

SCHWARZ THEORY 27 Algorithm 4.1 (Abstract Multiplicative Schwarz Method { Implementation Form) un+1 = MS(un; f) where the operation uNEW = MS(uOLD; f) is de ned as: Do k = 1; : : : ; J rk = IT k (f AuOLD) ek = Rkrk uNEW = uOLD + Ikek uOLD = uNEW End do. Note that the rst step through the loop in MS( ; ) gives: uNEW = uOLD + I1e1 = uOLD + I1R1IT 1 (f AuOLD) = (I I1R1IT 1 A)uOLD + I1R1IT 1 f: Continuing in this fashion, and by de ning Tk = IkRkIT k A, we see that after the full loop in MS( ; ) the solution transforms according to: un+1 = (I TJ )(I TJ 1) (I T1)un + Bf; where B is a quite complicated combination of the operators Rk, Ik, IT k , and A. By de ning Ek = (I Tk)(I Tk 1) (I T1), we see that Ek = (I Tk)Ek 1. Therefore, since Ek 1 = I Bk 1A for some (implicitly de ned) Bk 1, we can identify the operators Bk through the recursion Ek = I BkA = (I Tk)Ek 1, giving BkA = I (I Tk)Ek 1 = I (I Bk 1A) + Tk(I Bk 1A) = Bk 1A + Tk TkBk 1A = Bk 1A+ IkRkIT k A IkRkIT k ABk 1A = Bk 1 + IkRkIT k IkRkIT k ABk 1 A; so that Bk = Bk 1 + IkRkIT k IkRkIT k ABk 1. But this means the above algorithm is equivalent to: Algorithm 4.2 (Abstract Multiplicative Schwarz Method { Operator Form) un+1 = un + B(f Aun) = (I BA)un + Bf where the multiplicative Schwarz error propagator E is de ned by: E = I BA = (I TJ )(I TJ 1) (I T1); Tk = IkRkIT k A; k = 1; : : : ; J: The operator B BJ is de ned implicitly, and obeys the recursion: B1 = I1R1IT 1 ; Bk = Bk 1 + IkRkIT k IkRkIT k ABk 1; k = 2; : : : ; J: An abstract additive Schwarz method, employing corrections in the spaces Hk, has the form: Algorithm 4.3 (Abstract Additive Schwarz Method { Implementation Form) un+1 = MS(un; f) where the operation uNEW = MS(uOLD; f) is de ned as: r = f AuOLD Do k = 1; : : : ; J rk = IT k r ek = Rkrk uNEW = uOLD + Ikek End do. 28 ABSTRACT SCHWARZ THEORY Since each loop iteration depends only on the original approximation uOLD, we see that the full correction to the solution can be written as the sum: un+1 = un + B(f Aun) = un + J Xk=1 IkRkIT k (f Aun); where the preconditioner B has the form B = PJk=1 IkRkIT k , and the error propagator is E = I BA. Therefore, the above algorithm is equivalent to: Algorithm 4.4 (Abstract Additive Schwarz Method { Operator Form) un+1 = un + B(f Aun) = (I BA)un + Bf where the additive Schwarz error propagator E is de ned by: E = I BA = I J Xk=1Tk; Tk = IkRkIT k A; k = 1; : : : ; J: The operator B is de ned explicitly as B =PJk=1 IkRkIT k . 4.2 Subspace splitting theory We now consider the framework of x4.1, employing the abstract results of x3.4. First, we prove some simple results about projectors, and the relationships between the operators Rk on the spaces Hk and the resulting operators Tk = IkRkIT k A on the space H. We then consider the \splitting" of the space H into subspaces IkHk, and the veri cation of the assumptions required to apply the abstract theory of x3.4 is reduced to deriving an estimate of the \splitting constant". Recall that an orthogonal projector is an operator P 2 L(H;H) having a closed subspace V H as its range (on which P acts as the identity), and having the orthogonal complement of V, denoted as V? H, as its null space. By this de nition, the operator I P is also clearly a projector, but having the subspace V? as range and V as null space. In other words, a projector P splits a Hilbert space H into a direct sum of a closed subspace and its orthogonal complement as follows: H = V V? = PH (I P )H: The following lemma gives a useful characterization of a projection operator; note that this characterization is often used as an equivalent alternative de nition of a projection operator. Lemma 4.1 Let A 2 L(H;H) be SPD. Then the operator P 2 L(H;H) is an A-orthogonal projector if and only if P is A-self-adjoint and idempotent (P 2 = P ). Proof. See [24], Theorem 9.5-1, page 481. Lemma 4.2 Assume dim(Hk) dim(H), Ik : Hk 7! H, null(Ik) = f0g, and that A is SPD. Then Qk = Ik(IT k Ik) 1IT k ; Pk = Ik(IT k AIk) 1IT k A; are the unique orthogonal and A-orthogonal projectors onto IkHk. Proof. By assuming that null(Ik) = f0g, we guarantee that both null(IT k Ik) = f0g and null(IT k AIk) = f0g, so that both Qk and Pk are well-de ned. It is easily veri ed that Qk is self-adjoint and Pk is A-self-adjoint, and it is immediate that Q2k = Qk and that P 2 k = Pk. Clearly, Qk : H 7! IkHk, and Pk : H 7! IkHk, so that by Lemma 4.1 these operators are orthogonal and A-orthogonal projectors onto IkHk. All that remains is to show that these operators are unique. By de nition, a projector onto a subspace IkHk acts as the identity on IkHk, and as the zero operator on (IkHk)?. Therefore, any two projectors Pk and ~ Pk onto IkHk must act identically on the entire space H = IkHk (IkHk)?, and therefore Pk = ~ Pk. Similarly, Qk is unique. ABSTRACT SCHWARZ THEORY 29 We now make the following natural assumption regarding the operators Rk A 1 k . Assumption 4.1 The operators Rk 2 L(Hk;Hk) are SPD. Further, there exists a subspace Vk Hk, and parameters 0 < !0 !1 < 2, such that (a) !0(Akvk; vk) (AkRkAkvk; vk), 8vk 2 Vk Hk, k = 1; : : : ; J , (b) (AkRkAkvk; vk) !1(Akvk; vk), 8vk 2 Hk, k = 1; : : : ; J . This implies that on the subspace Vk Hk, it holds that 0 < !0 i(RkAk), k = 1; : : : ; J , whereas on the entire space Hk, it holds that i(RkAk) !1 < 2, k = 1; : : : ; J . There are several consequences of the above assumption which will be useful later. Lemma 4.3 Assumption 4.1(b) implies that 0 < i(RkAk) !1, and (I RkAk) = kI RkAkkAk < 1. Proof. Since R and A are SPD by assumption, we have by Lemma 2.6 that RA is A-SPD. By Assumption 4.1(b), the Rayleigh quotients are bounded above by !1, so that 0 < i(RA) !1: Thus, (I RA) = max i j i(I RA)j = max i j1 i(RA)j: Clearly then (I RA) < 1 since 0 < !1 < 2. Lemma 4.4 Assumption 4.1(b) implies that (Akvk; vk) !1(R 1 k vk; vk); 8vk 2 Hk. Proof. We drop the subscripts for ease of exposition. By Assumption 4.1(b), (ARAv; v) !1(Av; v), so that !1 bounds the Raleigh quotients generated by RA. Since RA is similar to R1=2AR1=2, we must also have that (R1=2AR1=2v; v) !1(v; v): But this implies (AR1=2v;R1=2v) !1(R 1R1=2v;R1=2v); or (Aw;w) !1(R 1w;w); 8w 2 H. Lemma 4.5 Assumption 4.1(b) implies that Tk = IkRkIT k A is A-self-adjoint and A-non-negative, and (Tk) = kTkkA !1 < 2: Proof. That Tk = IkRkIT k A is A-self-adjoint and A-non-negative follows immediately from the symmetry of Rk and Ak. To show the last result, we employ Lemma 4.4 to obtain (ATkv; Tkv) = (AIkRkIT k Av; IkRkIT k Av) = (IT k AIkRkIT k Av;RkIT k Av) = (AkRkIT k Av;RkIT k Av) !1(R 1 k RkIT k Av;RkIT k Av) = !1(IT k Av;RkIT k Av) = !1(AIkRkIT k Av; v) = !1(ATkv; v): Now, from the Schwarz inequality, we have (ATkv; Tkv) !1(ATkv; v) !1(ATkv; Tkv)1=2(Av; v)1=2; or that (ATkv; Tkv)1=2 !1(Av; v)1=2; which implies that kTkkA !1 < 2. The key idea in all of the following theory involves the splitting of the original Hilbert space H into a collection of subspaces IkVk IkHk H. It will be important for the splitting to be stable in a certain sense, which we state as the following assumption.SCHWARZ THEORY 29 We now make the following natural assumption regarding the operators Rk A 1 k . Assumption 4.1 The operators Rk 2 L(Hk;Hk) are SPD. Further, there exists a subspace Vk Hk, and parameters 0 < !0 !1 < 2, such that (a) !0(Akvk; vk) (AkRkAkvk; vk), 8vk 2 Vk Hk, k = 1; : : : ; J , (b) (AkRkAkvk; vk) !1(Akvk; vk), 8vk 2 Hk, k = 1; : : : ; J . This implies that on the subspace Vk Hk, it holds that 0 < !0 i(RkAk), k = 1; : : : ; J , whereas on the entire space Hk, it holds that i(RkAk) !1 < 2, k = 1; : : : ; J . There are several consequences of the above assumption which will be useful later. Lemma 4.3 Assumption 4.1(b) implies that 0 < i(RkAk) !1, and (I RkAk) = kI RkAkkAk < 1. Proof. Since R and A are SPD by assumption, we have by Lemma 2.6 that RA is A-SPD. By Assumption 4.1(b), the Rayleigh quotients are bounded above by !1, so that 0 < i(RA) !1: Thus, (I RA) = max i j i(I RA)j = max i j1 i(RA)j: Clearly then (I RA) < 1 since 0 < !1 < 2. Lemma 4.4 Assumption 4.1(b) implies that (Akvk; vk) !1(R 1 k vk; vk); 8vk 2 Hk. Proof. We drop the subscripts for ease of exposition. By Assumption 4.1(b), (ARAv; v) !1(Av; v), so that !1 bounds the Raleigh quotients generated by RA. Since RA is similar to R1=2AR1=2, we must also have that (R1=2AR1=2v; v) !1(v; v): But this implies (AR1=2v;R1=2v) !1(R 1R1=2v;R1=2v); or (Aw;w) !1(R 1w;w); 8w 2 H. Lemma 4.5 Assumption 4.1(b) implies that Tk = IkRkIT k A is A-self-adjoint and A-non-negative, and (Tk) = kTkkA !1 < 2: Proof. That Tk = IkRkIT k A is A-self-adjoint and A-non-negative follows immediately from the symmetry of Rk and Ak. To show the last result, we employ Lemma 4.4 to obtain (ATkv; Tkv) = (AIkRkIT k Av; IkRkIT k Av) = (IT k AIkRkIT k Av;RkIT k Av) = (AkRkIT k Av;RkIT k Av) !1(R 1 k RkIT k Av;RkIT k Av) = !1(IT k Av;RkIT k Av) = !1(AIkRkIT k Av; v) = !1(ATkv; v): Now, from the Schwarz inequality, we have (ATkv; Tkv) !1(ATkv; v) !1(ATkv; Tkv)1=2(Av; v)1=2; or that (ATkv; Tkv)1=2 !1(Av; v)1=2; which implies that kTkkA !1 < 2. The key idea in all of the following theory involves the splitting of the original Hilbert space H into a collection of subspaces IkVk IkHk H. It will be important for the splitting to be stable in a certain sense, which we state as the following assumption. 30 ABSTRACT SCHWARZ THEORY Assumption 4.2 Given any v 2 H =PJk=1 IkHk, IkHk H, there exists subspaces IkVk IkHk H = PJk=1 IkVk, and a particular splitting v =PJk=1 Ikvk, vk 2 Vk, such that J Xk=1kIkvkk2A S0kvk2A; for some splitting constant S0 > 0. The following key lemma (in the case of inclusion and projection as prolongation and restriction) is sometimes referred to as Lions' Lemma [25], although the multiple-subspace case is essentially due to Widlund [36]. Lemma 4.6 Under Assumption 4.2 it holds that 1 S0 kvk2A J Xk=1(APkv; v); 8v 2 H: Proof. Given any v 2 H, we employ the splitting of Assumption 4.2 to obtain kvk2A = J Xk=1(Av; Ikvk) = J Xk=1(IT k Av; vk) = J Xk=1(IT k A(Ik(IT k AIk) 1IT k A)v; vk) = J Xk=1(APkv; Ikvk): Now, let ~ Pk = (IT k AIk) 1IT k A, so that Pk = Ik ~ Pk. Then kvk2A = J Xk=1(IT k AIk ~ Pkv; vk) = J Xk=1(Ak ~ Pkv; vk) J Xk=1(Akvk; vk)1=2(Ak ~ Pkv; ~ Pkv)1=2 J Xk=1(Akvk; vk)!1=2 J Xk=1(Ak ~ Pkv; ~ Pkv)!1=2 = J Xk=1(AIkvk; Ikvk)!1=2 J X k=1(Ak ~ Pkv; ~ Pkv)!1=2 = J Xk=1kIkvkk2A!1=2 J Xk=1(Ak ~ Pkv; ~ Pkv)!1=2 S1=2 0 kvkA J X k=1(AIk ~ Pkv; Ik ~ Pkv)!1=2 = S1=2 0 kvkA J Xk=1(APkv; Pkv)!1=2 ; 8v 2 H: Since (APkv; Pkv) = (APkv; v), dividing the above by kvkA and squaring yields the result. The next intermediate result will be useful in the case that the subspace solver Rk is e ective on only the part of the subspace Hk, namely Vk Hk. Lemma 4.7 Under Assumptions 4.1(a) and 4.2 (for the same subspaces IkVk IkHk) it holds that J X k=1(R 1 k vk; vk) S0 !0 kvk2A; 8v = J Xk=1 Ikvk 2 H; vk 2 Vk Hk: Proof. With v =PJk=1 Ikvk, where we employ the splitting in Assumption 4.2, we have J Xk=1(R 1 k vk; vk) = J Xk=1(AkA 1 k R 1 k vk; vk) = J Xk=1(Akvk; vk) (AkA 1 k R 1 k vk; vk) (Akvk; vk) J Xk=1(Akvk; vk)max vk 6=0 (AkA 1 k R 1 k vk; vk) (Akvk; vk) J X k=1! 1 0 (Akvk; vk) = J Xk=1! 1 0 (AIkvk; Ikvk) = J Xk=1! 1 0 kIkvkk2A S0 !0 kvk2A; which proves the lemma. ABSTRACT SCHWARZ THEORY 31 The following lemma relates the constant appearing in the \splitting" Assumption 3.9 of the product and sum operator theory to the subspace splitting constant appearing in Assumption 4.2 above. Lemma 4.8 Under Assumptions 4.1(a) and 4.2 (for the same subspaces IkVk IkHk) it holds that kvk2A S0 !0 J Xk=1(ATkv; v); 8v 2 H: Proof. Given any v 2 H, we begin with the splitting in Assumption 4.2 as follows kvk2A = (Av; v) = J Xk=1(Av; Ikvk) = J Xk=1(IT k Av; vk) = J Xk=1(RkIT k Av;R 1 k vk): We employ now the Cauchy-Schwarz inequality in the Rk inner-product, yielding kvk2A J Xk=1(RkR 1 k vk; R 1 k vk)!1=2 J Xk=1(RkIT k Av; IT k Av)!1=2 S0 !0 1=2 kvkA J Xk=1(AIkRkIT k Av;Av)!1=2 = S0 !0 1=2 kvkA J Xk=1(ATkv; v)!1=2 ; where we have employed Lemma 4.7 for the last inequality. Dividing the inequality above by kvkA and squaring yields the lemma. In order to employ the product and sum theory, we must quantify the interaction of the operators Tk. As the Tk involve corrections in subspaces, we will see that the operator interaction properties will be determined completely by the interaction of the subspaces. Therefore, we introduce the following notions to quantify the interaction of the subspaces involved. De nition 4.2 (Strong interaction matrix) The interaction matrix 2 L(RJ;RJ) is de ned to have as entries ij the smallest constants satisfying: j(AIiui; Ijvj)j ij(AIiui; Iiui)1=2(AIjvj ; Ijvj)1=2; 1 i; j J; ui 2 Hi; vj 2 Hj: De nition 4.3 (Weak interaction matrix) The strictly upper-triangular interaction matrix 2 L(RJ;RJ) is de ned to have as entries ij the smallest constants satisfying: j(AIiui; Ijvj)j ij(AIiui; Iiui)1=2(AIjvj ; Ijvj)1=2; 1 i < j J; ui 2 Hi; vj 2 Vj Hj : The following lemma relates the interaction properties of the subspaces speci ed by the strong interaction matrix to the interaction properties of the associated subspace correction operators Tk = IkRkIT k A. Lemma 4.9 For the strong interaction matrix given in De nition 4.2, it holds that j(ATiu; Tjv)j ij(ATiu; Tiu)1=2(ATjv; Tjv)1=2; 1 i; j J; 8u; v 2 H: Proof. Since Tku = IkRkIT k Au = Ikuk, where uk = RkIT k Au, the lemma follows simply from the de nition of in De nition 4.2 above. Remark 4.12. Note that the weak interaction matrix in De nition 4.3 involves a subspace Vk Hk, which will be necessary in the analysis of multigrid-like methods. Unfortunately, this will preclude the simple application of the product operator theory of the previous sections. In particular, we cannot estimate the constant C2 required for the use of Corollary 3.6, because we cannot show Lemma 3.15 for arbitrary Tk. In order to prove Lemma 3.15, we would need to employ the upper-triangular portion of the strong interaction matrix in De nition 4.2, involving the entire space Hk, which is now di erent from the upper-triangular weak interaction matrix (employing only the subspace Vk) de ned as above in De nition 4.3. There was no such distinction between the weak and strong interaction matrices in the product and sum operator theory of the previous sections; the weak interaction matrix was de ned simply as the strictly upper-triangular portion of the strong interaction matrix.SCHWARZ THEORY 31 The following lemma relates the constant appearing in the \splitting" Assumption 3.9 of the product and sum operator theory to the subspace splitting constant appearing in Assumption 4.2 above. Lemma 4.8 Under Assumptions 4.1(a) and 4.2 (for the same subspaces IkVk IkHk) it holds that kvk2A S0 !0 J Xk=1(ATkv; v); 8v 2 H: Proof. Given any v 2 H, we begin with the splitting in Assumption 4.2 as follows kvk2A = (Av; v) = J Xk=1(Av; Ikvk) = J Xk=1(IT k Av; vk) = J Xk=1(RkIT k Av;R 1 k vk): We employ now the Cauchy-Schwarz inequality in the Rk inner-product, yielding kvk2A J Xk=1(RkR 1 k vk; R 1 k vk)!1=2 J Xk=1(RkIT k Av; IT k Av)!1=2 S0 !0 1=2 kvkA J Xk=1(AIkRkIT k Av;Av)!1=2 = S0 !0 1=2 kvkA J Xk=1(ATkv; v)!1=2 ; where we have employed Lemma 4.7 for the last inequality. Dividing the inequality above by kvkA and squaring yields the lemma. In order to employ the product and sum theory, we must quantify the interaction of the operators Tk. As the Tk involve corrections in subspaces, we will see that the operator interaction properties will be determined completely by the interaction of the subspaces. Therefore, we introduce the following notions to quantify the interaction of the subspaces involved. De nition 4.2 (Strong interaction matrix) The interaction matrix 2 L(RJ;RJ) is de ned to have as entries ij the smallest constants satisfying: j(AIiui; Ijvj)j ij(AIiui; Iiui)1=2(AIjvj ; Ijvj)1=2; 1 i; j J; ui 2 Hi; vj 2 Hj: De nition 4.3 (Weak interaction matrix) The strictly upper-triangular interaction matrix 2 L(RJ;RJ) is de ned to have as entries ij the smallest constants satisfying: j(AIiui; Ijvj)j ij(AIiui; Iiui)1=2(AIjvj ; Ijvj)1=2; 1 i < j J; ui 2 Hi; vj 2 Vj Hj : The following lemma relates the interaction properties of the subspaces speci ed by the strong interaction matrix to the interaction properties of the associated subspace correction operators Tk = IkRkIT k A. Lemma 4.9 For the strong interaction matrix given in De nition 4.2, it holds that j(ATiu; Tjv)j ij(ATiu; Tiu)1=2(ATjv; Tjv)1=2; 1 i; j J; 8u; v 2 H: Proof. Since Tku = IkRkIT k Au = Ikuk, where uk = RkIT k Au, the lemma follows simply from the de nition of in De nition 4.2 above. Remark 4.12. Note that the weak interaction matrix in De nition 4.3 involves a subspace Vk Hk, which will be necessary in the analysis of multigrid-like methods. Unfortunately, this will preclude the simple application of the product operator theory of the previous sections. In particular, we cannot estimate the constant C2 required for the use of Corollary 3.6, because we cannot show Lemma 3.15 for arbitrary Tk. In order to prove Lemma 3.15, we would need to employ the upper-triangular portion of the strong interaction matrix in De nition 4.2, involving the entire space Hk, which is now di erent from the upper-triangular weak interaction matrix (employing only the subspace Vk) de ned as above in De nition 4.3. There was no such distinction between the weak and strong interaction matrices in the product and sum operator theory of the previous sections; the weak interaction matrix was de ned simply as the strictly upper-triangular portion of the strong interaction matrix. 32 ABSTRACT SCHWARZ THEORY We can, however, employ the original Theorem 3.5 by attempting to estimate C1 directly, rather than employing Corollary 3.6 and estimating C1 indirectly through C0 and C2. The following result will allow us to do this, and still employ the weak interaction property above in De nition 4.3. Lemma 4.10 Under Assumptions 4.1 and 4.2 (for the same subspaces IkVk IkHk), it holds that kvk2A S0 !0 [1 + !1k k2]2 J Xk=1(ATkEk 1v;Ek 1v); 8v 2 H; where is the weak interaction matrix of De nition 4.3. Proof. We employ the splitting of Assumption 4.2, namely v =PJk=1 Ikvk, vk 2 Vk Hk, as follows: kvk2A = J Xk=1(Av; Ikvk) = J X k=1(AEk 1v; Ikvk) + J Xk=1(A[I Ek 1]v; Ikvk) = J X k=1(AEk 1v; Ikvk) + J Xk=1 k 1 Xi=1(ATiEi 1v; Ikvk) = S1 + S2: We now estimate S1 and S2 separately. For the rst term, we have: S1 = J Xk=1(AEk 1v; Ikvk) = J Xk=1(IT k AEk 1v; vk) = J Xk=1(RkIT k AEk 1v;R 1 k vk) J Xk=1(RkIT k AEk 1v; IT k AEk 1v)1=2(R 1 k vk; vk)1=2 = J X k=1(ATkEk 1v;Ek 1v)1=2(R 1 k vk; vk)1=2 J Xk=1(ATkEk 1v;Ek 1v)!1=2 J X k=1(R 1 k vk; vk)!1=2 : where we have employed the Cauchy-Schwarz inequality in the Rk inner-product for the rst inequality and in RJ for the second. Employing now Lemma 4.7 (requiring Assumptions 4.1 and 4.2) to bound the right-most term, we have S1 S0 !0 1=2 kvkA J Xk=1(ATkEk 1v;Ek 1v)!1=2 : We now bound the term S2, employing the weak interaction matrix given in De nition 4.3 above, as follows: S2 = J Xk=1 k 1 Xi=1(ATiEi 1v; Ikvk) = J X k=1 k 1 Xi=1(AIi[RiIT i AEi 1v]; Ikvk) J Xk=1 J Xi=1 ikkIi[RiIT i AEi 1v]kAkIkvkkA = J Xk=1 J Xi=1 ikkTiEi 1vkAkIkvkkA = ( x;y)2; where x;y 2 RJ, xk = kIkvkkA, yi = kTiEi 1vkA, and ( ; )2 is the usual Euclidean inner-product in RJ. Now, we have that S2 ( x;y)2 k k2kxk2kyk2 = k k2 J Xk=1(ATkEk 1v; TkEk 1v)!1=2 J Xk=1(AIkvk; Ikvk)!1=2 !1=2 1 k k2 J Xk=1(ATkEk 1v;Ek 1v)!1=2 J Xk=1(Akvk; vk)!1=2 ; ABSTRACT SCHWARZ THEORY 33 since Ak = IT k AIk, and by Lemma 3.2, which may be applied because of Lemma 4.5. By Lemma 4.4, we have (Akvk; vk) !1(R 1 k vk; vk), and employing this result along with Lemma 4.7 gives S2 !1k k2 J X k=1(ATkEk 1v;Ek 1v)!1=2 J X k=1(R 1 k vk; vk)!1=2 S0 !0 1=2 kvkA!1k k2 J Xk=1(ATkEk 1v;Ek 1v)!1=2 : Combining the two results gives nally kvk2A S1 + S2 S0 !0 1=2 kvkA [1 + !1k k2] J X k=1(ATkEk 1v;Ek 1v)!1=2 ; 8v 2 H: Dividing by kvkA and squaring yieldings the result. Remark 4.13. Although our language and notation is quite di erent, the proof we have given above for Lemma 4.10 is similar to results in [41] and [17]. Similar ideas and results appear [35]. The main ideas and techniques underlying proofs of this type were originally developed in [7, 8, 39]. 4.3 Product and sum splitting theory for non-nested Schwarz methods The main theory for Schwarz methods based on non-nested subspaces, as in the case of overlapping domain decomposition-like methods, may be summarized in the following way. We still consider an abstract method, but we assume it satis es certain assumptions common to real overlapping Schwarz domain decomposition methods. In particular, due to the local nature of the operators Tk for k 6= 0 arising from subspaces associated with overlapping subdomains, it will be important to allow for a special global operator T0 for global communication of information (the need for T0 will be demonstrated later). Therefore, we use the analysis framework of the previous sections which includes the use of a special global operator T0. Note that the local nature of the remaining Tk will imply that ( ) Nc, where Nc is the number of maximum number of subdomains which overlap any subdomain in the region. The analysis of domain decomposition-type algorithms is in most respects a straightforward application of the theory of products and sums of operators, as presented earlier. The theory for multigrid-type algorithms is more subtle; we will discuss this in the next section. Let the operators E and P be de ned as: E = (I TJ )(I TJ 1) (I T0); (21) P = T0 + T1 + + TJ ; (22) where the operators Tk 2 L(H;H) are de ned in terms of the approximate corrections in the spaces Hk as: Tk = IkRkIT k A; k = 0; : : : ; J; (23) where Ik : Hk 7! H; null(Ik) = f0g; IkHk H; H = J Xk=1 IkHk: The following assumptions are required; note that the following theory employs many of the assumptions and lemmas of the previous sections, for the case that Vk Hk. Assumption 4.3 (Subspace solvers) The operators Rk 2 L(Hk;Hk) are SPD. Further, there exists parameters 0 < !0 !1 < 2, such that !0(Akvk; vk) (AkRkAkvk; vk) !1(Akvk; vk); 8vk 2 Hk; k = 0; : : : ; J:SCHWARZ THEORY 33 since Ak = IT k AIk, and by Lemma 3.2, which may be applied because of Lemma 4.5. By Lemma 4.4, we have (Akvk; vk) !1(R 1 k vk; vk), and employing this result along with Lemma 4.7 gives S2 !1k k2 J X k=1(ATkEk 1v;Ek 1v)!1=2 J X k=1(R 1 k vk; vk)!1=2 S0 !0 1=2 kvkA!1k k2 J Xk=1(ATkEk 1v;Ek 1v)!1=2 : Combining the two results gives nally kvk2A S1 + S2 S0 !0 1=2 kvkA [1 + !1k k2] J X k=1(ATkEk 1v;Ek 1v)!1=2 ; 8v 2 H: Dividing by kvkA and squaring yieldings the result. Remark 4.13. Although our language and notation is quite di erent, the proof we have given above for Lemma 4.10 is similar to results in [41] and [17]. Similar ideas and results appear [35]. The main ideas and techniques underlying proofs of this type were originally developed in [7, 8, 39]. 4.3 Product and sum splitting theory for non-nested Schwarz methods The main theory for Schwarz methods based on non-nested subspaces, as in the case of overlapping domain decomposition-like methods, may be summarized in the following way. We still consider an abstract method, but we assume it satis es certain assumptions common to real overlapping Schwarz domain decomposition methods. In particular, due to the local nature of the operators Tk for k 6= 0 arising from subspaces associated with overlapping subdomains, it will be important to allow for a special global operator T0 for global communication of information (the need for T0 will be demonstrated later). Therefore, we use the analysis framework of the previous sections which includes the use of a special global operator T0. Note that the local nature of the remaining Tk will imply that ( ) Nc, where Nc is the number of maximum number of subdomains which overlap any subdomain in the region. The analysis of domain decomposition-type algorithms is in most respects a straightforward application of the theory of products and sums of operators, as presented earlier. The theory for multigrid-type algorithms is more subtle; we will discuss this in the next section. Let the operators E and P be de ned as: E = (I TJ )(I TJ 1) (I T0); (21) P = T0 + T1 + + TJ ; (22) where the operators Tk 2 L(H;H) are de ned in terms of the approximate corrections in the spaces Hk as: Tk = IkRkIT k A; k = 0; : : : ; J; (23) where Ik : Hk 7! H; null(Ik) = f0g; IkHk H; H = J Xk=1 IkHk: The following assumptions are required; note that the following theory employs many of the assumptions and lemmas of the previous sections, for the case that Vk Hk. Assumption 4.3 (Subspace solvers) The operators Rk 2 L(Hk;Hk) are SPD. Further, there exists parameters 0 < !0 !1 < 2, such that !0(Akvk; vk) (AkRkAkvk; vk) !1(Akvk; vk); 8vk 2 Hk; k = 0; : : : ; J: 34 ABSTRACT SCHWARZ THEORY Assumption 4.4 (Splitting constant) Given any v 2 H, there exists S0 > 0 and a particular splitting v =PJk=0 Ikvk, vk 2 Hk, such that J Xk=0kIkvkk2A S0kvk2A: De nition 4.4 (Interaction matrix) The interaction matrix 2 L(RJ;RJ) is de ned to have as entries ij the smallest constants satisfying: j(AIiui; Ijvj)j ij(AIiui; Iiui)1=2(AIjvj ; Ijvj)1=2; 1 i; j J; ui 2 Hi; vj 2 Hj: Theorem 4.11 (Multiplicative method) Under Assumptions 4.3 and 4.4, it holds that kEk2A 1 !0(2 !1) S0(6 + 6!2 1 ( )2) : Proof. By Lemma 4.5, Assumption 4.3 implies that Assumption 3.8 holds, with ! = !1. By Lemma 4.8, we know that Assumptions 4.3 and 4.4 imply that Assumption 3.9 holds, with C0 = S0=!0. By Lemma 4.9, we know that De nition 4.4 is equivalent to De nition 3.2 for . Therefore, the theorem follows by application of Theorem 3.23. Theorem 4.12 (Additive method) Under Assumptions 4.3 and 4.4, it holds that A(P ) S0( ( ) + 1)!1 !0 : Proof. By Lemma 4.5, Assumption 4.3 implies that Assumption 3.8 holds, with ! = !1. By Lemma 4.8, we know that Assumptions 4.3 and 4.4 imply that Assumption 3.9 holds, with C0 = S0=!0. By Lemma 4.9, we know that De nition 4.4 is equivalent to De nition 3.2 for . Therefore, the theorem follows by application of Theorem 3.24. Remark 4.14. Note that Assumption 4.3 is equivalent to A(RkAk) !1 !0 ; k = 0; : : : ; J; or maxkf A(RkAk)g !1=!0. Thus, the result in Theorem 4.12 can be written as: A(P ) S0( ( ) + 1)max k f A(RkAk)g: Therefore, the global condition number is completely determined by the local condition numbers, the splitting constant, and the interaction property. Remark 4.15. We have the default estimate for ( ): ( ) J: For use of the theory above, we must also estimate the splitting constant S0, and the subspace solver spectral bounds !0 and !1, for each particular application. Remark 4.16. Note that if a coarse space operator T0 is not present, then the alternate bounds from the previous sections could have been employed. However, the advantage of the above approach is that the additional space H0 does not adversely e ect the bounds, while it provides an additional space to help satisfy the splitting assumption. In fact, in the nite element case, it is exactly this coarse space which allows one to show that S0 does not depend on the number of subspaces, yielding optimal algorithms when a coarse space is involved. Remark 4.17. The theory in this section was derived mainly from work in the domain decomposition community, due chie y to Widlund and his co-workers. In particular, our presentation owes much to [39] and [14]. ABSTRACT SCHWARZ THEORY 35 4.4 Product and sum splitting theory for nested Schwarz methods The main theory for Schwarz methods based on nested subspaces, as in the case of multigrid-like methods, is summarized in this section. By \nested" subspaces, we mean here that there are additional subspaces Vk Hk of importance, and we re ne the analysis to consider these addition nested subspaces Vk. Of course, we must still assume thatPJk=1 IkVk = H. Later, when analyzing multigrid methods, we will consider in fact a nested sequence I1H1 I2H2 HJ H, with Vk Hk, although this assumption is not necessary here. We will however assume here that one space H1 automatically performs the role of a \global" space, and hence it will not be necessary to include a special global space H0 as in the non-nested case. Therefore, we will employ the analysis framework of the previous sections which does not speci cally include a special global operator T0. (By working with the subspaces Vk rather than the Hk we will be able to avoid the problems encountered with a global operator interacting with all other operators, as in the previous sections.) The analysis of multigrid-type algorithms is more subtle than analysis for overlapping domain decomposition methods, in that the e ciency of the method comes from the e ectiveness of simple linear methods (e.g., Gauss-Seidel iteration) at reducing the error in a certain sub-subspace Vk of the \current" space Hk. The overall e ect on the error is not important; just the e ectiveness of the linear method on error subspace Vk. The error in the remaining space HknVk is handled by subspace solvers in the other subspaces, since we assume that H = PJk=1 IkVk. Therefore, in the analysis of the nested space methods to follow, the spaces Vk Hk introduced earlier will play a key role. This is in contrast to the non-nested theory of the previous section, where it was taken to be the case that Vk Hk. Roughly speaking, nested space algorithms \split" the error into components in Vk, and if the subspace solvers in each space Hk are good at reducing the error in Vk, then the overall method will be good. Let the operators E and P be de ned as: E = (I TJ )(I TJ 1) (I T1); (24) P = T1 + T2 + + TJ ; (25) where the operators Tk 2 L(H;H) are de ned in terms of the approximate corrections in the spaces Hk as: Tk = IkRkIT k A; k = 1; : : : ; J; (26) where Ik : Hk 7! H; null(Ik) = f0g; IkHk H; H = J Xk=1 IkHk: The following assumptions are required. Assumption 4.5 (Subspace solvers) The operators Rk 2 L(Hk;Hk) are SPD. Further, there exists subspaces IkVk IkHk H =PJk=1 IkVk, and parameters 0 < !0 !1 < 2, such that !0(Akvk; vk) (AkRkAkvk; vk); 8vk 2 Vk Hk; k = 1; : : : ; J; (AkRkAkvk; vk) !1(Akvk; vk); 8vk 2 Hk; k = 1; : : : ; J: Assumption 4.6 (Splitting constant) Given any v 2 H, there exists subspaces IkVk IkHk H = PJk=1 IkVk (the same subspaces Vk as in Assumption 4.5 above) and a particular splitting v = PJk=1 Ikvk, vk 2 Vk, such that J X k=1 kIkvkk2A S0kvk2A; 8v 2 H; for some splitting constant S0 > 0. De nition 4.5 (Strong interaction matrix) The interaction matrix 2 L(RJ;RJ) is de ned to have as entries ij the smallest constants satisfying: j(AIiui; Ijvj)j ij(AIiui; Iiui)1=2(AIjvj ; Ijvj)1=2; 1 i; j J; ui 2 Hi; vj 2 Hj:SCHWARZ THEORY 35 4.4 Product and sum splitting theory for nested Schwarz methods The main theory for Schwarz methods based on nested subspaces, as in the case of multigrid-like methods, is summarized in this section. By \nested" subspaces, we mean here that there are additional subspaces Vk Hk of importance, and we re ne the analysis to consider these addition nested subspaces Vk. Of course, we must still assume thatPJk=1 IkVk = H. Later, when analyzing multigrid methods, we will consider in fact a nested sequence I1H1 I2H2 HJ H, with Vk Hk, although this assumption is not necessary here. We will however assume here that one space H1 automatically performs the role of a \global" space, and hence it will not be necessary to include a special global space H0 as in the non-nested case. Therefore, we will employ the analysis framework of the previous sections which does not speci cally include a special global operator T0. (By working with the subspaces Vk rather than the Hk we will be able to avoid the problems encountered with a global operator interacting with all other operators, as in the previous sections.) The analysis of multigrid-type algorithms is more subtle than analysis for overlapping domain decomposition methods, in that the e ciency of the method comes from the e ectiveness of simple linear methods (e.g., Gauss-Seidel iteration) at reducing the error in a certain sub-subspace Vk of the \current" space Hk. The overall e ect on the error is not important; just the e ectiveness of the linear method on error subspace Vk. The error in the remaining space HknVk is handled by subspace solvers in the other subspaces, since we assume that H = PJk=1 IkVk. Therefore, in the analysis of the nested space methods to follow, the spaces Vk Hk introduced earlier will play a key role. This is in contrast to the non-nested theory of the previous section, where it was taken to be the case that Vk Hk. Roughly speaking, nested space algorithms \split" the error into components in Vk, and if the subspace solvers in each space Hk are good at reducing the error in Vk, then the overall method will be good. Let the operators E and P be de ned as: E = (I TJ )(I TJ 1) (I T1); (24) P = T1 + T2 + + TJ ; (25) where the operators Tk 2 L(H;H) are de ned in terms of the approximate corrections in the spaces Hk as: Tk = IkRkIT k A; k = 1; : : : ; J; (26) where Ik : Hk 7! H; null(Ik) = f0g; IkHk H; H = J Xk=1 IkHk: The following assumptions are required. Assumption 4.5 (Subspace solvers) The operators Rk 2 L(Hk;Hk) are SPD. Further, there exists subspaces IkVk IkHk H =PJk=1 IkVk, and parameters 0 < !0 !1 < 2, such that !0(Akvk; vk) (AkRkAkvk; vk); 8vk 2 Vk Hk; k = 1; : : : ; J; (AkRkAkvk; vk) !1(Akvk; vk); 8vk 2 Hk; k = 1; : : : ; J: Assumption 4.6 (Splitting constant) Given any v 2 H, there exists subspaces IkVk IkHk H = PJk=1 IkVk (the same subspaces Vk as in Assumption 4.5 above) and a particular splitting v = PJk=1 Ikvk, vk 2 Vk, such that J X k=1 kIkvkk2A S0kvk2A; 8v 2 H; for some splitting constant S0 > 0. De nition 4.5 (Strong interaction matrix) The interaction matrix 2 L(RJ;RJ) is de ned to have as entries ij the smallest constants satisfying: j(AIiui; Ijvj)j ij(AIiui; Iiui)1=2(AIjvj ; Ijvj)1=2; 1 i; j J; ui 2 Hi; vj 2 Hj: 36 ABSTRACT SCHWARZ THEORY De nition 4.6 (Weak interaction matrix) The strictly upper-triangular interaction matrix 2 L(RJ;RJ) is de ned to have as entries ij the smallest constants satisfying: j(AIiui; Ijvj)j ij(AIiui; Iiui)1=2(AIjvj ; Ijvj)1=2; 1 i < j J; ui 2 Hi; vj 2 Vj Hj : Theorem 4.13 (Multiplicative method) Under Assumptions 4.5 and 4.6, it holds that kEk2A 1 !0(2 !1) S0(1 + !1k k2)2 : Proof. The proof of this result is more subtle than the additive method, and requires more work than a simple application of the product operator theory. This is due to the fact that the weak interaction matrix of De nition 4.6 speci cally involves the subspace Vk Hk. Therefore, rather than employing Theorem 3.25, which employs Corollary 3.6 indirectly, we must do a more detailed analysis, and employ the original Theorem 3.5 directly. (See the remarks preceding Lemma 4.10.) By Lemma 4.5, Assumption 4.5 implies that Assumption 3.1 holds, with ! = !1. Now, to employ Theorem 3.5, it su ces to realize that Assumption 3.3 holds with with C1 = S0(1 + !1k k2)2=!0. This follows from Lemma 4.10. Theorem 4.14 (Additive method) Under Assumptions 4.5 and 4.6, it holds that A(P ) S0 ( )!1 !0 : Proof. By Lemma 4.5, Assumption 4.5 implies that Assumption 3.8 holds, with ! = !1. By Lemma 4.8, we know that Assumptions 4.5 and 4.6 imply that Assumption 3.9 holds, with C0 = S0=!0. By Lemma 4.9, we know that De nition 4.5 is equivalent to De nition 3.2 for . Therefore, the theorem follows by application of Theorem 3.26. Remark 4.18. We have the default estimates for k k2 and ( ): k k2 pJ(J 1)=2 < J; ( ) J: For use of the theory above, we must also estimate the splitting constant S0, and the subspace solver spectral bounds !0 and !1, for each particular application. Remark 4.19. The theory in this section was derived from several sources; in particular, our presentation owes much to [39], [17], and to [41]. 5. Applications to domain decomposition Domain decomposition methods were rst proposed by H.A. Schwarz as a theoretical tool for studying elliptic problems on complicated domains, constructed as the union of simple domains. An interesting early reference not often mentioned is [21], containing both analysis and numerical examples, and references to the original work by Schwarz. In this section, we brie y describe the fundamental overlapping domain decomposition methods, and apply the theory of the previous sections to give convergence rate bounds. 5.1 Variational formulation and subdomain-based subspaces Given a domain and coarse triangulation by J regions f kg of mesh size Hk, we re ne (several times) to obtain a ne mesh of size hk. The regions de ned by the initial triangulation k are then extended by k to form the \overlapping subdomains" 0k. Now, let V and V0 denote the nite element spaces associated with the hk and Hk triangulation of , respectively. The variational problem in V has the form: Find u 2 V such that a(u; v) = f(v); 8v 2 V: The form a( ; ) is bilinear, symmetric, coercive, and bounded, whereas f( ) is linear and bounded. Therefore, through the Riesz representation theorem we can associate with the above problem an abstract operator equation Au = f , where A is SPD. Domain decomposition methods can be seen as iterative methods for solving the above operator equation, involving approximate projections of the error onto subspaces of V associated with the overlapping subdomains 0k. To be more speci c, let Vk = H1 0 ( 0k) \ V , k = 1; : : : ; J ; it is not di cult to show that V = V1 + + VJ , where the coarse space V0 may also be included in the sum. 5.2 The multiplicative and additive Schwarz methods We denote as Ak the restriction of the operator A to the space Vk, corresponding to (any) discretization of the original problem restricted to the subdomain 0k. Algebraically, it can be shown that Ak = IT k AIk, where Ik is the natural inclusion in H and IT k is the corresponding projection. The property that Ik is the natural inclusion and IT k is the corresponding projection holds if either Vk is a nite element space or the Euclidean space Rnk (in the case of multigrid, Ik and IT k are inclusion and projection only in the nite element space case). In other words, domain decomposition methods automatically satisfy the variational condition, De nition 4.1, in the subspaces Vk, k 6= 0, for any discretization method. Now, if Rk A 1 k , we can de ne the approximate A-orthogonal projector from V onto Vk as Tk = IkRkIT k A. An overlapping domain decomposition method can be written as the basic linear method, Algorithm 2.1, where the multiplicative Schwarz error propagator E is: E = (I TJ )(I TJ 1) (I T0): The additive Schwarz preconditioned system operator P is: P = T0 + T1 + + TJ : Therefore, the overlapping multiplicative and additive domain decomposition methods t exactly into the framework of abstract multiplicative and additive Schwarz methods discussed in the previous sections. 5.3 Algebraic domain decomposition methods As remarked above, for domain decomposition methods it automatically holds that Ak = IT k AIk, where Ik is the natural inclusion, IT k is the corresponding projection, and Vk is either a nite element space orRnk. While this variational condition holds for multigrid methods only in the case of nite element discretizations, or when directly enforced as in algebraic multigrid methods (see the next section), the condition holds naturally and automatically for domain decomposition methods employing any discretization technique. 37 38 APPLICATIONS TO DOMAIN DECOMPOSITION We see that the Schwarz method framework then applies equally well to domain decomposition methods based on other discretization techniques (box-method or nite di erences), or to algebraic equations having a block-structure which can be viewed as being associated with the discretization of an elliptic equation over a domain. The Schwarz framework can be used to provide a convergence analysis even in the algebraic case, although the results may be suboptimal compared to the nite element case when more information is available about the continuous problem. 5.4 Convergence theory for the algebraic case For domain decomposition methods, the local nature of the projection operators will allow for a simple analysis of the interaction properties required for the Schwarz theory. To quantify the local nature of the projection operators, assume that we are given H = PJk=0 IkHk along with the subspaces IkHk H, and denote as Pk the A-orthogonal projector onto IkHk. We now make the following de nition. De nition 5.1 For each operator Pk, 1 k J , de ne N (k) c to be the number of operators Pi such that PkPi 6= 0, 1 i J , and let Nc = max1 k JfN (k) c g. Remark 5.20. This is a natural condition for domain decomposition methods, where N (k) c represents the number of subdomains which overlap a given domain associated with Pk, excluding a possible coarse space I0H0. By treating the projector P0 separately in the analysis, we allow for a global space H0 which may in fact interact with all of the other spaces. Note that Nc J in general with Schwarz methods; with domain decomposition, we can show that Nc = O(1). Our use of the notation Nc comes from the idea that Nc represents essentially the minimum number of colors required to color the subdomains so that no two subdomains sharing interior mesh points have the same color. (If the domains were non-overlapping, then this would be a case of the four-color problem, so that in two dimensions it would always hold that Nc 4.) The following splitting is the basis for applying the theory of the previous sections. Note that this splitting is well-de ned in a completely algebraic setting without further assumptions. Lemma 5.1 Given any v 2 H =PJk=0 IkHk, IkHk H, there exists a particular splitting v =PJk=0 Ikvk, vk 2 Hk, such that J Xk=0kIkvkk2A S0kvk2A; for the splitting constant S0 =PJk=0 kQkk2A. Proof. Let Qk 2 L(H;Hk) be the orthgonal projectors onto the subspaces Hk. We have that Hk = QkH, and any v 2 H can be represented uniquely as v = J Xk=0Qkv = J Xk=0 Ikvk; vk 2 Hk: We have then that J Xk=0kIkvkk2A = J Xk=0kQkvk2A J Xk=0kQkk2Akvk2A = S0kvk2A; where S0 =PJk=0 kQkk2A. Lemma 5.2 It holds that ( ) Nc. Proof. This follows easily, since ( ) k k1 = maxjfPi j ijjg Nc. We make the following assumption on the subspace solvers. Assumption 5.1 Assume there exists SPD operators Rk 2 L(Hk;Hk) and parameters 0 < !0 !1 < 2, such that !0(Akvk; vk) (AkRkAkvk; vk) !1(Akvk; vk); 8vk 2 Hk; k = 1; : : : ; J: APPLICATIONS TO DOMAIN DECOMPOSITION 39 Theorem 5.3 Under Assumption 5.1, the multiplicative Schwarz domain decomposition method has an error propagator which satis es: kEk2A 1 !0(2 !1) S0(6 + 6!2 1N2 c ) : Proof. By Assumption 5.1, we have that Assumption 4.3 holds. By Lemma 5.1, we have that Assumption 4.4 holds, with S0 = PJk=0 kQkk2A. By Lemma 5.2, we have that for as in De nition 4.4, it holds that ( ) Nc. The proof now follows from Theorem 4.11. Theorem 5.4 Under Assumption 5.1, the additive Schwarz domain decomposition method as a preconditioner gives a condition number bounded by: A(P ) S0(1 +Nc)!1 !0 : Proof. By Assumption 5.1, we have that Assumption 4.3 holds. By Lemma 5.1, we have that Assumption 4.4 holds, with S0 = PJk=0 kQkk2A. By Lemma 5.2, we have that for as in De nition 4.4, it holds that ( ) Nc. The proof now follows from Theorem 4.12. 5.5 Improved results through nite element theory If a coarse space is employed, and the overlap of the subdomains k is on the order of the subdomain size Hk, i.e., k = cHk, then one can bound the splitting constant S0 to be independent of the mesh size and the number of subdomains J . Required to prove such a result is some elliptic regularity or smoothness on the solution to the original continuous problem: Find u 2 H1 0( ) such that a(u; v) = (f; v); 8v 2 H1 0 ( ): The regularity assumption is stated as an apriori estimate or regularity inequality of the following form: The solution to the continuous problem satis es u 2 H1+ ( ) for some real number > 0, and there exists a constant C such that kukH1+ ( ) CkfkH 1( ): If this regularity inequality holds for the continuous solution, one can show the following result by employing some results from interpolation theory and nite element approximation theory. Lemma 5.5 There exists a splitting v =PJk=0 Ikvk, vk 2 Hk such that J Xk=0kIkvkk2A S0kvk2A; 8v 2 H; where S0 is independent of J (and hk and Hk). Proof. Refer for example to the proof in [39] and the references therein to related results. 6. Applications to multigrid Multigrid methods were rst developed by Federenko in the early 1960's, and have been extensively studied and developed since they became widely known in the late 1970's. In this section, we brie y describe the linear multigrid method as a Schwarz method, and apply the theory of the previous sections to give convergence rate bounds. 6.1 Recursive multigrid and nested subspaces Consider a set of nite-dimensional Hilbert spaces Hk of increasing dimension: dim(H1) < dim(H2) < < dim(HJ ): The spaces Hk, which may for example be nite element function spaces, or simply Rnk (where nk = dim(Hk)), are assumed to be connected by prolongation operators Ik k 1 2 L(Hk 1;Hk), and restriction operators Ik 1 k 2 L(Hk;Hk 1). We can use these various operators to de ne mappings Ik that provide a nesting structure for the set of spaces Hk as follows: I1H1 I2H2 IJHJ H; where IJ = I; Ik = IJ J 1IJ 1 J 2 Ik+2 k+1Ik+1 k ; k = 1; : : : ; J 1: We assume that each space Hk is equipped with an inner-product ( ; )k inducing the norm k kk = ( ; )1=2 k . Also associated with each Hk is an operator Ak, assumed to be SPD with respect to ( ; )k. It is assumed that the operators satisfy variational conditions: Ak 1 = Ik 1 k AkIk k 1; Ik 1 k = (Ik k 1)T : (27) These conditions hold naturally in the nite element setting, and are imposed directly in algebraic multigrid methods. Given B A 1 in the space H, the basic linear method constructed from the preconditioned system BAu = Bf has the form: un+1 = un BAun + Bf = (I BA)un + Bf: (28) Now, given some B, or some procedure for applying B, we can either formulate a linear method using E = I BA, or employ a CG method for BAu = Bf if B is SPD. 6.2 Variational multigrid as a multiplicative Schwarz method The recursive formulation of multigrid methods has been well-known for more than fteen years; mathematically equivalent forms of the method involving product error propagators have been recognized and exploited theoretically only very recently. In particular, it can be shown [7, 20, 29] that if the variational conditions (27) hold, then the multigrid error propagator can be factored as: E = I BA = (I TJ )(I TJ 1) (I T1); (29) where: IJ = I; Ik = IJ J 1IJ 1 J 2 Ik+2 k+1Ik+1 k ; k = 1; : : : ; J 1; (30) T1 = I1A 1 1 IT 1 A; Tk = IkRkIT k A; k = 2; : : : ; J; (31) where Rk A 1 k is the \smoothing" operator employed in each space Hk. It is not di cult to show that with the de nition of Ik in equation (30), the variational conditions (27) imply that additional variational conditions hold between the nest space and each of the subspaces separately, as required for the Schwarz theory: Ak = IT k AIk: (32) 40 APPLICATIONS TO MULTIGRID 41 6.3 Algebraic multigrid methods Equations arising in various application areas often contain complicated discontinuous coe cients, the shapes of which may not be resolvable on all coarse mesh element boundaries as required for accurate nite element approximation (and as required for validity of nite element error estimates). Multigrid methods typically perform badly, and even the regularity-free multigrid convergence theory [7] is invalid. Possible approaches include coe cient averaging methods (cf. [1]) and the explicit enforcement of the conditions (27) (cf. [1, 11, 33]). By introducing a symbolic stencil calculus and employing MAPLE or MATHEMATICA, the conditions (27) can be enforced algebraically in an e cient way for certain types of sparse matrices; details may be found for example in the appendix of [20]. If one imposes the variational conditions (27) algebraically, then from our comments in the previous section we know that algebraic multigrid methods can be viewed as multiplicative Schwarz methods, and we can attempt to analyze the convergence rate of algebraic multigrid methods using the Schwarz theory framework. 6.4 Convergence theory for the algebraic case The following splitting is the basis for applying the theory of the previous sections. Note that this splitting is well-de ned in a completely algebraic setting without further assumptions. Lemma 6.1 Given any v 2 H = PJk=0 IkHk, Ik 1Hk 1 IkHk H, there exists subspaces IkVk IkHk H =PJk=1 IkVk, and a particular splitting v =PJk=0 Ikvk, vk 2 Vk, such that J Xk=0kIkvkk2A kvk2A: The subspaces are IkVk = (Pk Pk 1)H, and the splitting is v =PJk=1(Pk Pk 1)v. Proof. We have the projectors Pk : H 7! IkHk as de ned in Lemma 4.2, where we take the convention that PJ = I, and that P0 = 0. Since Ik 1Hk 1 IkHk, we know that PkPk 1 = Pk 1Pk = Pk 1. Now, let us de ne: P̂1 = P1; P̂k = Pk Pk 1; k = 2; : : : ; J: By Theorem 9.6-2 in [24] we have that each P̂k is a projection. (It is easily veri ed that P̂k is idempotent and A-self-adjoint.) De ne now IkVk = P̂kH = (Pk Pk 1)H = (IkA 1 k IT k A Ik 1A 1 k 1IT k 1A)H = Ik(A 1 k Ik k 1A 1 k 1(Ik k 1)T )IT k AH; k = 1; : : : ; J; where we have used the fact that two forms of variational conditions hold, namely those of equation (27) and equation (32). Note that P̂kP̂j = (Pk Pk 1)(Pj Pj 1) = PkPj PkPj 1 Pk 1Pj + Pk 1Pj 1: Thus, if k > j, then P̂kP̂j = Pj Pj 1 Pj + Pj 1 = 0: Similarly, if k < j, then P̂kP̂j = Pk Pk Pk 1 + Pk 1 = 0: Thus, H = I1V1 I2V2 IJVJ = P̂1H P̂2H P̂JH; and P =PJk=1 P̂k = I de nes a splitting (an A-orthogonal splitting) of H. We then have that kvk2A = (APv; v) = J Xk=1(AP̂kv; v) = J X k=1(AP̂kv; P̂kv) = J X k=1 kP̂kvk2A = J Xk=1 kIkvkk2A: 42 APPLICATIONS TO MULTIGRID For the particular splitting employed above, the weak interaction property is quite simple. Lemma 6.2 The (strictly upper-triangular) interaction matrix 2 L(RJ;RJ), having entries ij as the smallest constants satisfying: j(AIiui; Ijvj)j ij(AIiui; Iiui)1=2(AIjvj ; Ijvj)1=2; 1 i < j J; ui 2 Hi; vj 2 Vj Hj ; satis es 0 for the subspace splitting IkVk = P̂kH = (Pk Pk 1)H. Proof. Since P̂jPi = (Pj Pj 1)Pi = PjPi Pj 1Pi = Pi Pi = 0 for i < j, we have that IjVj = P̂jH is orthogonal to IiHi = PiH, for i < j. Thus, it holds that (AIiui; Ijvj) = 0; 1 i < j J; ui 2 Hi; vj 2 Vj Hj : The most di cult assumption to verify will be the following one. Assumption 6.1 There exists SPD operators Rk and parameters 0 < !0 !1 < 2 such that !0(Akvk; vk) (AkRkAkvk; vk); 8vk 2 Vk; IkVk = (Pk Pk 1)H IkHk; k = 1; : : : ; J; (AkRkAkvk; vk) !1(Akvk; vk); 8vk 2 Hk; k = 1; : : : ; J: With this single assumption, we can state the main theorem. Theorem 6.3 Under Assumption 6.1, the multigrid method has an error propagator which satis es: kEk2A 1 !0(2 !1): Proof. By Assumption 6.1, Assumption 4.5 holds. The splitting in Lemma 6.1 shows that Assumption 4.6 holds, with S0 = 1. Lemma 6.2 shows that for as in De nition 4.6, it holds that 0. The theorem now follows by Theorem 4.13. Remark 6.21. In order to analyze the convergence rate of an algebraic multigridmethod, we now see that we must be able to estimate the two parameters !0 and !1 in Assumption 6.1. However, in an algebraic multigrid method, we are free to choose the prolongation operator Ik, which of course also in uences Ak = IT k AIk. Thus, we can attempt to select the prolongation operator Ik and the subspace solver Rk together, so that Assumption 6.1 will hold, independent of the number of levels J employed. In other words, the Schwarz theory framework can be used to help design an e ective algebraic multigrid method. Whether it will be possible to select Rk and Ik satisfying the above requirements is the subject of future work. 6.5 Improved results through nite element theory It can be shown that Assumption 6.1 holds for parameters !0 and !1 independent of the mesh size and number of levels J , if one assumes some elliptic regularity or smoothness on the solution to the original continuous problem: Find u 2 H1 0( ) such that a(u; v) = (f; v); 8v 2 H1 0 ( ): This regularity assumption is stated as an apriori estimate or regularity inequality of the following form: The solution to the continuous problem satis es u 2 H1+ ( ) for some real number > 0, and there exists a constant C such that kukH1+ ( ) CkfkH 1( ): If this regularity inequality holds with = 1 for the continuous solution, one can show the following result by employing some results from interpolation theory and nite element approximation theory. Lemma 6.4 There exists SPD operators Rk and parameters 0 < !0 !1 < 2 such that !0(Akvk; vk) (AkRkAkvk; vk); 8vk 2 Vk; IkVk = (Pk Pk 1)H IkHk; k = 1; : : : ; J; (AkRkAkvk; vk) !1(Akvk; vk); 8vk 2 Hk; k = 1; : : : ; J: APPLICATIONS TO MULTIGRID 43 Proof. See for example the proof in [41]. More generally, assume only that u 2 H1( ) (so that the regularity inequality holds only with = 0), and that there exists L2( )-like orthogonal projectors Qk onto the nite element spaces Mk, where we take the convention that QJ = I and Q0 = 0. This de nes the splitting v = J Xk=1(Qk Qk 1)v; which is central to the BPWX theory [7]. Employing this splitting along with results from nite element approximation theory, it is shown in [7], using a similar Schwarz theory framework, that kEk2A 1 C J1+ ; 2 f0; 1g: This result holds even in the presence of coe cient discontinuities (the constants being independent of the jumps in the coe cients). 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