Sharp condition number estimates for the symmetric -Lagrange multiplier method Stephen W. Drury and Sébastien Loisel Abstract Domain decomposition methods are used to find the numerical solution of large boundary value problems in parallel. In optimized domain decomposition methods, one solves a Robin subproblem on each subdomain, where the Robin parameter a must be tuned (or optimized) for good performance. We show that the -Lagrange multiplier method can be analyzed using matrix analytical techniques and we produce sharp condition number estimates. Introduction. Consider the model problem u = f in Ω and u = 0 on Ω, () where Ω is the domain, f is a given forcing and u H0 (Ω) is the unknown solution. In the present paper, we describe a symmetric -Lagrange multiplier (SLM) domain decomposition method to solve elliptic problems such as (). When we discretize () using e.g. piecewise linear finite elements, we obtain a linear system of the form Au = f, () where u R n is the finite element coefficient vector of the approximation to the solution u of (). Dept. of Mathematics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A K6, drury@math.mcgill.ca Dept. of Mathematics, Heriot-Watt University, Edinburgh EH4 4AS, United Kingdom, S.Loisel@hw.ac.uk
Stephen W. Drury and Sébastien Loisel We now consider the domain decomposition [Toselli and Widlund, 005] Ω = Γ Ω... Ω p,whereω,...,ω p arethe (open, disjoint) subdomains and Γ = Ω p k= Ω k is the artificial interface. We introduce the local problems u k = f in Ω k, (PDE) u k = 0 on Ω k Ω, (natural b.c.) (3) (a+d ν )u k = λ k on Ω k Γ, (artificial b.c.) where a > 0 is the Robin tuning parameter and k =,...,p and D ν denotes the directional derivative in the outwards pointing normal ν of Ω k. The interface Γ is artificial in that it is not a natural part of the physical problem () but instead is introduced purely for the purpose of calculation. We again discretize the systems (3) using a finite element method. The Robin b.c. in (3) gives rise to a mass matrix on the interface Γ Ω k, which is spectrally equivalent to ai. Hence, after a suitable mild change of basis, we obtain the discrete system [ AIIk A IΓk A ΓIk A ΓΓk +ai u k ] { [ }} ] { uik = u Γk f k [ {}} ] { [ ] fik 0 +. (4) f Γk λ k The FETI-LM algorithm was introduced in [Farhat et al., 000] for cases without cross-points, while the general case including cross points was introduced and analyzed in [Loisel, 0a]. The method consists of finding the value of λ = [λ T,...,λT p ]T which yields solutions u,...,u p to (4) in such a way that u,...,u p meet continuously across Γ and glue together into the unique solution u of (). The main result of the present paper is a new estimate the condition number of FETI-LM algorithms using matrix analytical techniques. This new idea produces sharp condition number estimates with much more straightforward proof techniques than the techniques used in [Loisel, 0a] (where the estimates are not sharp). As a result, the present paper is a logical follow-up to [Loisel, 0a]. The present paper focuses on -level algorithms which are known not to scale. Scalable algorithms are considered in [Loisel, 0b] and [Drury and Loisel, 0]. Our paper is organized as follows. In Section, we give the symmetric - Lagrange multiplier method for general domains with cross points. In Section 3, we give spectral estimates including our main result on the condition number of the symmetric -Lagrange multiplier system. in Section 4, we verify this Theorem with some numerical experiments.
Sharp condition estimates of SLM 3 The symmetric -Lagrange multiplier method. We now describe the -Lagrange multiplier method that we analyze in the present paper. Consider the local problems (4) and eliminate the interior degrees of freedom to obtain the relation a where u G { }} { u. u p = Q {}} { { }} { a(s +ai)... a(s +ai) g g. g p + λ { }} { λ. λ p, (5) S k = A ΓΓk A ΓIk A IIk A IΓk and g k = f Γk A ΓIk A IIk f Ik are the Dirichlet-to-Neumann maps and accumulated right-hand-sides. The matrices S k are symmetric and semidefinite. Since Q = a(s +ai), we find that the spectrum σ(q) is contained in the set [ǫ, ǫ] {} for some ǫ > 0. The eigenvalue of Q comes from the kernel of S and hence the kernel of Q I is spanned by the indicating functions of the subdomains that float. We define E to be the orthogonal projection onto the kernel of Q I.. Relations between (4) and () and continuity. We define the boolean restriction matrix R k by selecting rows of the n n identity matrix corresponding to those vertices of Ω that are in Ω k Ω. As a result, from a finite element coefficient vector v corresponding to a finite element function v H0(Ω), we can define a finite element coefficient vector v k = R k v, which corresponds to a finite element function v H (Ω k ) H0(Ω), which is obtained by restricting v to Ω k. The identity Ω = p k= Ω k induces the following relations between (4) and (): A = k= R T k [ ] AIIk A IΓk R A ΓIk A k and f = ΓΓk f k. (6) Each interface vertex x i Γ is adjacent to m i subdomains. As a result, the many-sided trace u G defined by (5) contains m i entries corresponding to x i, one per subdomain adjacent to x i. We define the orthogonal projection matrix K which averages function values for each interface vertex x i. A many-sided trace u G corresponds to local functions u,...,u p that k=
4 Stephen W. Drury and Sébastien Loisel meet continuously across Γ if and only if Ku G = u G. (7). A problem in λ. The symmetric -Lagrange multiplier (SLM) system is given by (Q K)λ = Qg. (8) We further let E be the orthogonal projection onto the kernel of Q I. Lemma. Assume that EK <. The problem () is equivalent to (8). Proof. In order to solve () using local problems (4), one should find Robin boundary values λ,...,λ p which result in local solutions u,...,u p that meet continuously across Γ. As a result, we impose the condition (7), which we multiply by a > 0 and convert to an expression in λ using (5) to obtain Ka(S +ai) (λ+g) = a(s +ai) (λ+g) or (I K)Qλ = (K I)Qg (9) With this continuity condition, there is clearly a unique u which restricts to the u j : u j = R j u, j =,...,p. (0) Imposing continuity is not sufficient, we must also ensure that the fluxes match. Indeed, if we impose on the solution u of (0) that the equation () should hold, one obtains f = Au (6) = (4),(6) = f j= j= R T Γj A NjR Γj u (0) = R T j ( ) 0 λ j au Γj RΓj T A Nju j () j= () Canceling the f terms on each side and multiplying by K, we obtain Kλ Kau G = 0. Using (5), we obtain K(Q I)λ = KQg. (3) We add (9) and (3) to obtain (8). To see that the solution of (8) is unique, observe that the ranges of E and K intersect trivially by the hypothesis that EK <. As a result, the
Sharp condition estimates of SLM 5 eigenspace of Q of eigenvalue intersects trivially with the range of K and Q K is nonsingular. We will further discuss the choice of the parameter a in Section 3.. 3 Spectral estimates. If we use GMRES or MINRES on the symmetric indefinite system (8), the residual norm can be estimated as a function of the condition number of Q K, cf. [Driscoll et al., 998]. In order to estimate the condition number of Q K, we begin by giving a canonical form for the pair of projections E and K. Lemma. Let E and K be orthogonal projections. There is a choice of orthonormal basis that block diagonalizes E and K simultaneously and such that the blocks E k and K k of E and K satisfy E k { [ ]} 0 0,, 0 0 and K k { 0,, [ c k c k s k c k s k s k ]}, (4) where c k = cosθ k > 0, s k = sinθ k > 0 and θ k (0,π/) is a principal angle relating E and K. The canonical form(4) can be obtained from the CS decomposition[davis and Kahan, 969] by starting from E = diag(i,0) and picking orthonormal bases for the range and kernel of K. Due to space constraints, we omit this argument. We also give a technical lemma which describes the spectrum of a sum of certain symmetric matrices. Lemma 3. Let X, Y be symmetric matrices of dimensions m m. Let 0 < y min < y max and assume that σ(y) [y min,y max ]. Denote by ρ(x) the spectral radius of X and assume that ρ(x) < y min. Then, σ(x +Y) [y min ρ(x),y max +ρ(x)]. (5) Proof. This follows from a Theorem of Weyl [Horn and Johnson, 990, Theorem 4.3., pp 8 8]. 3. Condition number of Q K. We now come to our main result.
6 Stephen W. Drury and Sébastien Loisel Theorem. Let ǫ > 0. Assume that σ(q) [ǫ, ǫ] {}. Let E,K be orthogonal projections and assume that EK <. Then we have the sharp estimates [ ǫ+ ] (+ǫ) σ(q K) 4 EK ǫ,, and (6) κ(q K) ǫ+ (+ǫ) 4 EK ǫ = O(( EK ) ǫ ). (7) Proof. Let X = Q I ǫe and Y = I+ǫE K.Then, Q K = X+Y and we are in a position to use Lemma 3. We now estimate the spectral properties of X and Y. Spectral properties of X: Recall that E projects onto the eigenspace of Q with eigenvalue. As a result, after some orthonormal change of basis, we find that Q = diag(q 0,I) and E = diag(0,i) and hence ρ(x) ǫ. (8) Spectral properties of Y: Lemma shows that E and K block diagonalize simultaneously and Y is also block diagonal in the same basis. Using (4), we find that the kth block Y k of Y is given by if E k = K k = 0, if E k = 0, K k =, Y k = [ +ǫ ] if E k =, K k = 0, (9) +ǫ c k c ks k c k s k s k where the case E k = K k = is excluded by the hypothesis that EK <. As a result, the eigenvalues of Y k are in the set {±, +ǫ,λ ±(c k )}, where ; λ ± (c k ) = ǫ± (+ǫ) 4c k ǫ. (0) Note that EK = ρ(eke) = c k and that the functions λ ± (c k ) are monotonic in c k. Hence, we find the following bounds for the modulus of an eigenvalue of Y: σ(y) y min [ { }}{ y max {}}{ ] (+ǫ) 4 EK ǫ ǫ, +ǫ. ()
Sharp condition estimates of SLM 7 0 5 0 4 0 3 κ 0 0 0 0 0 5 0 4 0 3 0 0 0 0 EK 0 7 0 6 0 5 0 4 κ 0 3 0 0 0 0 0 5 0 4 0 3 0 0 0 0 ε Fig. Comparing random Q K (points) versus the estimate (7) (solid). Top: ǫ = 0., varying EK, 3000 repetitions. Bottom: EK = 0.99, varying ǫ, 3000 repetitions. Combining (5), (8) and () gives (6). The sharpness of the[ estimate is shown by considering the example Q = c diag(, ǫ) and K = c ] c c c c for c = 0 and c = EK.
8 Stephen W. Drury and Sébastien Loisel In view of Theorem, the Robin parameter a should be chosen so as to make ǫ as large as possible. This occurs precisely when a is the geometric mean of the extremal positive eigenvalues of S. More details can be found in [Loisel, 0a]. 4 Numerical verification. We verify numerically the validity of Theorem by generating random 5 5 matrices Q and E as follows. We set Q = diag(ǫ,q, ǫ,,) where q is chosen randomly between ǫ and ǫ. We generate randomly a -dimensional space and set K to be the orthogonalprojectiononto that space.we compare the resulting condition number κ = κ(q K) against (7), cf. Fig.. References Chandler Davis and W. M. Kahan. Some new bounds on perturbation of subspaces. Bulletin of the American Mathematical Society, pages 863 868, 969. Tobin A. Driscoll, Kim-Chuan Toh, and Lloyd N. Trefethen. From potential theory to matrix iterations in six steps. SIAM Review, pages 547 578, 998. Stephen W. Drury and Sébastien Loisel. The performance of optimized Schwarz and -Lagrange multiplier preconditioners for GMRES. Manuscript, 0. Charbel Farhat, Antonini Macedo, Michel Lesoinne, Francois-Xavier Roux, Frédéric Magoulès, and Armel de La Bourdonnaie. Two-level domain decomposition methods with lagrange multipliers for the fast iterative solution of acoustic scattering problems. Computer Methods in Applied Mechanics and Engineering, 84:3 39, 000. Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge University Press, 990. Sébastien Loisel. Condition number estimates for the nonoverlapping optimized Schwarz method and the -Lagrange multiplier method for general domains and cross points. Submitted to SIAM Journal on Numerical Analysis, 0a. Sébastien Loisel. Condition number estimates and weak scaling for -level -Lagrange multiplier methods for general domains and cross points. Submitted, 0b. Andrea Toselli and Olof B. Widlund. Domain Decomposition Methods Algorithms and Theory, volume 34 of Springer Series in Computational Mathematics. Springer Berlin Heidelberg, 005.