H. A. Schwarz. Gesammelte Mathematische Abhandlungen, volume 2, pages 133 143. Springer, 1890. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zfirich, volume 15, 1870, pp. 272 286. 19  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to subregions, and a problem with a coarse mesh which provides global transportation of information, are solved. These subproblems can often be solved in parallel. Therefore these methods are very promising for parallel computation. The Schwarz alternating method, proposed by H. A. Schwarz [68] in 1869, is the first domain decomposition algorithm. This method involves solving the boundary value problems on two subdomains sequentially. Recent work by P. L. Lions [46] has interpreted the method for two subdomains cases in a variational framework. It is not ideally suitable for parallel ....

....is symmetric and tridiagonal. See Parlett [64] for a complete discussion of the Lanczos method. 17 Chapter 2 Domain Decomposition Methods 2.1 Multiplicative Schwarz Methods The earliest domain decomposition method is the Schwarz alternating method, which was proposed by H. A. Schwarz in 1869, [68]. The convergence of this method was established by using the maximum principle. In the 1930 s, Sobolev extended the result to the partial differential equations of linear elasticity [74] We consider the domain, shown in Figure 2.1, with Omega = Omega 1 [ Omega 2 on which we wish to solve ( ....

H. A. Schwarz, Gesammelte Mathematische Abhandlungen, vol. 2, Springer, Berlin, 1890, pp. 133--143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....restriction of the original problem to subregions, or to a problem with a coarse mesh, are solved. The number of subproblems can be large and these methods are promising for parallel computation. The Schwarz alternating method, the first domain decomposition algorithm, was proposed by H.A. Schwarz [42] in 1869. It involves solving problems on two subdomains sequentially. In [34] P. L. Lions interpreted the method in a variational framework. Many subdomain cases were also considered. Because of the sequential nature of the method, it is not ideal for parallel computations. In [24] Dryja and ....

....to construct such preconditioners. Other preconditioners are also used in practices, e.g. the diagonal scaling and incomplete Cholesky preconditioners. 2.2 Multiplicative Schwarz Schemes The Schwarz alternating method is the first domain decomposition method, proposed by H. A. Schwarz in 1869, [42]. Schwarz established convergence using the maximum principle. In the 1930 s, Sobolev extended the result to the partial differential equations of linear elasticity [43] New theoretical tools for the analysis of multiplicative Schwarz schemes have been developed more recently. The continuous ....

H. A. Schwarz, Gesammelte Mathematische Abhandlungen, vol. 2, Springer, Berlin, 1890, pp. 133--143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....framework, known as the abstract Schwarz theory. This approach has been adopted by many authors; cf. 25, 55, 56] For a comprehensive treatment, see the monograph by Smith, Bjrstad, and Gropp [62] 2. 1 The classical Schwarz alternating method This method was introduced by Schwarz in 1870 [58] and is the oldest domain decomposition method. Let Omega = Omega 1 [ Omega 2 be a plane domain. The goal is to find the solution u to the problem Gamma Deltau = f in Omega u = 0 on Omega To this end, we compute a sequence u n that converges to u. We choose an initial guess u 0 . Then, ....

H. A. Schwarz. Gesammelte Mathematische Abhandlungen, volume 2, pages 133-- 143. Springer, Berlin, 1890. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286. 106

....Bramble, Pasciak, Wang, and Xu [7] and Xu [56] have extended the general framework making a systematic study of multiplicative Schwarz methods possible. The multiplicative algorithms are direct generalizations of the original alternating method discovered more than 120 years ago by H.A. Schwarz [46]. For other current projects, which also use the Schwarz framework, see Dryja, Smith, and Widlund [27] and Dryja and Widlund [32,33] When a two level method is used, the restrictions of the discrete elliptic problem to overlapping subregions, into which the given region has been decomposed, are ....

....Poisson s equation, to homogeneous Dirichlet conditions and to continuous, piecewise linear finite elements and a polygonal region Omega Gamma It is well known that the resulting space V h ae H 1 0( Omega Gamma ; i.e. it is conforming. For the problem considered in a 1870 paper by H.A. Schwarz [46], two overlapping subregions, Omega 0 1 and Omega 0 2 are used; the union of the two is Omega : There are two sequential, fractional steps of the iteration in which the approximate solution of the elliptic equation on Omega is updated by solving the given problem restricted to the ....

H. A. Schwarz. Gesammelte Mathematische Abhandlungen, volume 2, pages 133--143. Springer, Berlin, 1890. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....by Bramble, Pasciak, Wang and Xu [4] and Xu [36] have extended the general framework making a systematic study of multiplicative Schwarz methods possible. The multiplicative algorithms are direct generalizations of the original alternating method discovered more than 120 years ago by H. A. Schwarz [30]. We note that most of the work in recent years has focused on the positive definite, symmetric case. While this theory is quite general, the applications so far have primarily been to the solution of the often large linear systems of algebraic equations, which arise in the finite element ....

H. A. Schwarz, Gesammelte Mathematische Abhandlungen, vol. 2, Springer, Berlin, 1890, pp. 133--143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....Remark 2.5. The PSC method as presented here is similar to the so called additive Schwarz method. The terminology additive Schwarz method was attributed to Dryja and Widlund [37] and it reflects the fact that the method studied in [37] is a variant of an alternating algorithm proposed by Schwarz [68] in 1870. The terminology parallel subspace correction method was introduced by Xu [81] to more directly reflect, within the general framework studied in [81] the nature of this type of method (including not only DD methods but other methods such as multigrid methods and even the classic Jacobi ....

H. Schwarz, Gesammelte Mathematische Abhandlungen, Springer, Berlin, 2 (1890), pp. 133-- 143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, 15 (1870), pp. 272--286.

....H is the diameter of a subdomain and h is the diameter of an element. These algorithms are often accelerated using the conjugate gradient method. The particular domain decomposition algorithm is then defined by the preconditioner for the conjugate gradient method. The alternating Schwarz method [75] is believed to be the first domain decomposition algorithm. In its original form it involves solving problems on two subdomains serially. We discuss this algorithm more completely in Chapter 2. Domain decomposition algorithms where the subproblems are solved in such a serial fashion like the ....

....should chose the preconditioner M so that M Gamma1 A is well conditioned and the operation M Gamma1 x is inexpensive. 2.2 Multiplicative Schemes 2.2. 1 The Classical Alternating Schwarz Scheme The earliest domain decomposition method was suggested in an existence proof by Schwarz in 1869, [75]. Let us consider the domain as shown in Figure 2.1 with Omega 1 Omega 2 Figure 2.1: Subdomains for Schwarz s Method Omega = Omega 1 [ Omega 2 on which we wish to solve Gamma4u = f in Omega ; u = 0 on Omega : The Schwarz alternating procedure approximates the solution ....

H. A. Schwarz. Gesammelete Mathematische Abhandlungen, volume 2, pages 133--143. Springer, Berlin, 1890. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....In fact, the additive method that we describe below may not even converge if used as a stand alone iteration, and is therefore always used together with an outer iterative Krylov space method. The ASM method was originally motivated by the famous alternating iteration of H.A. Schwarz [39], but it can also be viewed as a generalization of the block Jacobi iteration [26] Define a restriction matrix R i by u i = R i u where u i is a grid function defined in the local data structure i. Here u is the global reservoir grid function and R i consists of zeros and ones in such a way that ....

H. A. Schwarz, Gesammelte Mathematische Abhandlungen, vol. 2, Springer, Berlin, 1890, pp. 133--143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....both levels in Model 6 gives identical condition numbers to the first case considered. This is expected since two first level substructures that are decoupled will remain decoupled at all higher levels. 3. 3 Additive Schwarz methods Domain decomposition preconditioners based on ideas from Schwarz [26], has received much attention, see [16] 17] and the references in these papers. Until recently, most of this work related to model problems and scalar elliptic equations. In his thesis Barry Smith [27] developed and tested a method for two and three dimensional linear elasticity. He proved the ....

H. A. Schwarz, Gesammelte Mathematische Abhandlungen, vol. 2, Springer, Berlin, 1890, pp. 133--143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp.272--286.

....here is in fact essentially the same as the so called additive Schwarz method with inexact local solvers. The terminology additive Schwarz method was attributed to Dryja and Widlund [32] and it reflects the fact that this method is a variant of an alternating algorithm proposed by Schwarz [59] in 1870. The terminology parallel subspace correction method was given in Xu [71] to more directly reflect the nature of this type of methods within the general framework studied in [71] The basic idea of this type of methods can be found, for example, in Nepomnyaschikh [55] and Matsokin and ....

H. Schwarz. Gesammelte Mathematische Abhandlungen, volume 2, pages 133--143. Springer, Berlin, 1890. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....effort has focused on the case of many subregions; cf. e.g. Bramble et al. [8, 9, 5, 7, 6] Dryja [13, 12] Dryja et al. [14, 15] and Widlund [29] The general technique of domain splitting is not new. H. A. Schwarz introduced such a technique in a theoretical context more than hundred years ago [27]. Engineers, in particular within the field of structural analysis, have used the principle extensively, partly to ease the task of modeling complex structures, partly in order to organize the large computational tasks on computers with limited memory [2, 26] The subdomain solution technique has ....

H. A. Schwarz, Gesammelte Mathematische Abhandlungen, vol. 2, Springer, Berlin, 1890, pp. 133--143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....n k (1 Gamma c 2 A C 2 A ) n=2 kr 0 k: Chapter 2 Domain decomposition methods 2.1 The Classical Alternating Schwarz method This method is believed to be the oldest domain decomposition method. It was introduced by H. A. Schwarz in 1870 in the case of two overlapping subregions, see [74]. In order to find the solution u to the problem ( Gamma Deltau = f in Omega ; u = g on Omega ; on the plane region Omega = Omega 1 [ Omega 2 ; a sequence of functions fu n g converging to u is computed. The sequence is constructed in the following way. Choose an initial guess u 0 . ....

H. A. Schwarz. Gesammelte Mathematische Abhandlungen, volume 2, pages 133-- 143. Springer, Berlin, 1890. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....ordinary differential equation. The proof is given for two subdomains for simplicity, though it can be generalized to any number of subdomains. The method is based on the iterative method for the solution of classical boundary value problems for harmonic functions proposed by H. A. Schwarz [15] and has been successfully implemented in [4] 7] 8] and [10] In [3] 9] and [12] it was shown, using a variational approach, that the Schwarz alternating method converges. Here, it is shown that the Schwarz alternating method, combined with a sinc collocation method, converges at an ....

H. A. Schwarz. Gesammelte Mathematische Abhandlungen, volume 2, pages 133 143. Springer, 1890. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zfirich, volume 15, 1870, pp. 272 286. 19

....linear systems of equations arising from Galerkin finite element discretizations of elliptic partial differential equations. The new algorithms belong to the family of overlapping Schwarz methods which is a variant of the classical Schwarz alternating algorithm, introduced in 1870 by H. A. Schwarz [22] in an existence proof of elliptic boundary value problems defined in certain irregular regions. This family of methods has attracted much attention in the past few years as convenient and powerful in the solution of partial differential equations, see, especially on parallel machines. For ....

H. A. Schwarz, Gesammelte Mathematische Abhandlungen, vol. 2, Springer, Berlin, 1890, pp. 133-- 143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....decades. Two books by Smith, Bj rstad, and Gropp [43] and Quarteroni and Valli [40] have appeared recently, and almost yearly international conferences are being held; see [16, 8, 9, 17, 22, 39, 23, 18, 4, 28, 24, 46] The rst domain decomposition method is the alternating method of H. A. Schwarz [42]. At the core of that work is a proof that the iterative method converges in the maximum norm at a geometric rate. In his work in the 1980 s [25, 26] P. L. Lions analyzed the convergence of Schwarz methods using two di erent methods, a maximum principle and Hilbert spaces. He also established the ....

H. A. Schwarz. Gesammelte Mathematische Abhandlungen, volume 2, pages 133-143. Springer, Berlin, 1890. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272-286.

....these architectures. Domain decomposition methods are preconditioning techniques that introduce parallelism into the solution process together with a substantial reduction of the number of Krylov subspace iterations. The original idea of domain decomposition dates back to the work by H. A. Schwarz [28] more than 120 years ago. Using an iterative procedure he proved the existence of solution to the Poisson equation in the case of a domain composed from a square and a disk that overlapped. There are a large number of references related to overlapping Schwarz methods. Here we will only mention a ....

H. A. Schwarz, Gesammelte Mathematische Abhandlungen, vol. 2, Springer, Berlin, 1890, pp. 133--143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp.272--286.

....above for the PCG method, this estimate seems rather poor; it emphasizes the need for a good preconditioner A, so that the parameters in (2.3) fall within reasonable limits to make the method of practical value. 2.2 Multiplicative and Additive Schwarz Methods As early as 1869, H. A. Schwarz [105] used a domain decomposition idea to construct solutions to the Laplace problem for easily decomposable domains. PierreLouis Lions restated the algorithm in variational form, in a paper that provided a starting point for the revival of the interest in the method [72] We concern ourselves with ....

H. A. Schwarz. Gesammelte Mathematische Abhandlungen, volume 2, pages 133--143. Springer, Berlin, 1890. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....scalar and vector functions. In Chapter 2, we first review some iterative methods for positive definite linear system of equations. We then discuss recent results on Schwarz methods, which are regarded as generalizations of the first domain decomposition method proposed in 1869 by H. A. Schwarz [78]. In particular, a general abstract variational framework developed by Dryja and Widlund is usaed to analyze Schwarz methods in terms of subspaces and bilinear forms. In Chapter 3, we introduce the elliptic model problem and corresponding discrete system using the standard h version finite element ....

....and relatively small matrix Tm . Questions related to convergence of the extremal eigenvalues of Tm to those of BA are considered in [69] 2. 2 The Classical alternating Schwarz method It is believed that the first domain decomposition method was proposed by Hermann Amandus Schwarz in 1869 [78]. It was originally used to show the existence of the solution of an elliptic boundary value problem on domains that consists of the union of 19 simple overlapping subregions. In each subregion it is a priori known how to solve the elliptic problem. For instance, assume that we have two ....

H. A. Schwarz. Gesammelte Mathematische Abhandlungen, volume 2, pages 133-- 143. Springer, Berlin, 1890. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....operator. In the study of the local components of the preconditioners, we can draw on the extensive knowledge of substructuring methods for a few subdomains; cf. e.g. Bjrstad and Widlund [3] Throughout, we regard our methods as Schwarz methods, generalizations of the alternating method of Schwarz [52] discovered more than 120 years ago. Historically Schwarz methods have primarily been associated with a division of the region into overlapping subregions. In recent years, research on this classical method and its additive variants has been quite active; cf. e.g. Dryja and Widlund [23] 24] ....

H. A. Schwarz, Gesammelte Mathematische Abhandlungen, vol. 2, Springer, Berlin, 1890, pp. 133--143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....are methods where the global domain is partitioned into disjoint (nonoverlapping) pieces. More recently, the use of iterative methods for this class of problems, has been investigated, see [3] and [6] and the references given in those papers. In a famous paper more than hundred years ago, Schwarz [21] developed his domain decomposition algorithm, based on computing the solution of two overlapping subproblems in an alternating fashion. This algorithm is therefore quite sequential in its original form, and not necessarily the best candidate for a parallel implementation. More recently P. L. ....

H. A. Schwarz, Gesammelete Mathematische Abhandlungen, vol. 2, Springer, Berlin, 1890, pp. 133--143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp.272--286.

....[4] 5] 6] 7] 8] Dryja [14] 15] Dryja et al. 16] 17] and Widlund [25] However, in this note, we regress and consider only the case of two subregions. We show that a method recently developed by Chan and Resasco [11] 12] is identical to the classical Schwarz algorithm, cf. Schwarz [21], accelerated by the standard conjugate gradient method. We note that in a recent technical report, which we received after submitting this paper to SIAM, Chan and Goovaerts [10] give another description of our main result using a linear algebra framework similar to ours. Domain decomposition ....

....have to be extreme for this to be of any real importance. 3. Remarks on theory and other related methods. The convergence analysis of Schwarz s method can be carried out in a number of ways. Traditionally, the maximum principle is used to obtain an upper bound for the spectral radius; cf. Schwarz [21] or Lions [20] Lions [19] has also derived a different upper bound that depends on the geometry through the maximum norm of the gradient of a certain function used in a partition of unity. Our analysis above has shown that the spectrum of the error propagation operator (9) can be expressed ....

H. A. Schwarz, Gesammelete Mathematische Abhandlungen, Vol. 2, Springer-Verlag, Berlin, New York, 1890, pp.133-143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, Vol. 15 (1870), pp. 272-286.

.... In the study of the local components of the preconditioners, we can draw on the extensive knowledge of substructuring methods for a few subdomains; cf. e.g. Bj rstad and Widlund [3] Throughout, we regard our methods as Schwarz methods, generalizations of the alternating method of Schwarz [54] discovered more than 120 years ago. Historically Schwarz methods have primarily been associated with a division of the region into overlapping subregions. In recent years, research on this classical method and its additive variants has been quite active; cf. e.g. Dryja and Widlund [24] 25] ....

H. A. Schwarz, Gesammelte Mathematische Abhandlungen, vol. 2, Springer, Berlin, 1890, pp. 133--143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....ordinary differential equation. The proof is given for two subdomains for simplicity, though it can be generalized to any number of subdomains. The method is based on the iterative method for the solution of classical boundary value problems for harmonic functions proposed by H. A. Schwarz [15] and has been successfully implemented in [4] 7] 8] and [10] In [3] 9] and [12] it was shown, using a variational approach, that the Schwarz alternating method converges. Here, it is shown that the Schwarz alternating method, combined with a sinc collocation method, converges at an ....

H. A. Schwarz. Gesammelte Mathematische Abhandlungen, volume 2, pages 133--143. Springer, 1890. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

....methods for the solution of large, sparse, nonsymmetric and or indefinite linear systems of equations obtained by discretizing elliptic partial differential equations. This family of methods originates from the classical Schwarz alternating algorithm, introduced in 1870 by H. A. Schwarz [37] in an existence proof for elliptic boundary value problems defined in certain irregular regions. This method has attracted much attention as a convenient computational method for the solution of a large class of elliptic or parabolic equations, see e.g. 14, 38] especially on parallel machines, ....

H. A. Schwarz, Gesammelte Mathematische Abhandlungen, vol. 2, Springer, Berlin, 1890, pp. 133--143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, volume 15, 1870, pp. 272--286.

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H. A. Schwartz, Gesammelte Mathematische Abhandlungen, Vol. 2, Springer, Berlin, 1890, pp. 133-143. First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, vol. 115, 1870, pp. 272-286.

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