J H Bramble, J E Pasciak, and A H Shatz. An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp., 46:361--369, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the continuous problem. Establishing discrete extension theorems for some typical discretization schemes is an important, highly technical and delicate next step in developing the theory. For the diffusion equation, discrete extension theorems are known in many cases, e.g. for the FEM, see [22, 62, 49, 55, 2, 50, 33, 58]. Discrete analogs of extension theorems in the present paper are apparently not yet known. One interesting class of applications of extension theorems has been developed in [8, 38, 9, 14, 10, 17, 61, 1, 11, 4, 18, 19, 39, 3, 7, 5, 20, 21] where efficient preconditioned iterative methods for ....

J. H. Bramble, J. E. Pasciak, and A. H. Schatz. An iterative method for elliptic problems on regions partitioned into substructures. Mathematics of Computation, 46(174):361--369, 1986.

....one subdomain to another. For a general introduction to the IR methodology the reader is referred to [14,11,12] Several interface relaxation methods, considered form the domain decomposition viewpoint, like the Schwartz method, the Poincare Steklov method, the Schur complement, can be found in [1,2,18,19,23]. A review of a large collection of IR methods is presented in [13] The convergence of these schemes depends, as expected, on the di erential operator, on the geometry of the original domain, and on the geometry of the subdomains chosen. This makes the selection of optimum values for the ....

James H. Bramble, Joseph E. Pasciak, and Alfred H. Schatz. An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp., 46(173):361-369, 1986.

....of ways [19, 57, 63] Some studies partition the computational mesh into local subregions. Solution methods are applied to each subregion independently to precondition the linear system associated with the large globally defined problem or to achieve a fast solution method through parallelization [15, 16, 17]. Other approaches couple the grid adaptivity to the iterative solution process. Local refinement is used to increase accuracy in the approximate solution and to speed the reduction of the residual in certain sub regions of the problem domain [63] The multilevel iteration method we use is based ....

J.H. Bramble, j.E. Pasciak, and A.H. Schatz. An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp., 46(174):361-369, 1986.

....above equivalence to construct multilevel preconditioners for the linear system for the Lagrange multipliers. Then the mixed method solutions oe h and u h are recovered via these multipliers. The construction of multilevel preconditioners for the mixed methods is inspired by the fundamental work [7], 22] where new systematic representations for preconditioners in the Neumann Dirichlet domain decomposition methods for conforming finite elements were suggested. The multilevel domain decomposition versions of these methods were outlined in detail in [23] 24] and their multigrid versions ....

J.H. Bramble, J.E. Pasciak, and A.H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp., 46 (1986), 361--369.

....0 : 0 0 . SEkEk 0 0 : 0 S V V 1 C C C C A : One may expect that the condition number of the preconditioned system, M Gamma1 1 S, is dependent on the discretization size h as well as on the size of the subdomains H . This is in fact the case as shown in [BPS86], Theorem 3. There exists a constant C independent of H and h, such that cond(M Gamma1 1 S) CH Gamma2 (1 ln(H=h) The other preconditioner which was also introduced in [BPS86] can be written in the form M 2 = 0 B B B B SE1E1 0 : 0 0 . SEkEk 0 0 : ....

....on the discretization size h as well as on the size of the subdomains H . This is in fact the case as shown in [BPS86] Theorem 3. There exists a constant C independent of H and h, such that cond(M Gamma1 1 S) CH Gamma2 (1 ln(H=h) The other preconditioner which was also introduced in [BPS86] can be written in the form M 2 = 0 B B B B SE1E1 0 : 0 0 . SEkEk 0 0 : 0 A V 1 C C C C A : Here A V is a matrix resulting from the discretization of the problem only on vertices V . Global coupling between the vertices, introduced by the matrix A V , ....

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Bramble J. H., Pasciak J. E., and Shatz A. H. (1986) An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp. 46: 361--369.

....This constraint will be incorporated by an augmented Lagrangian technique of [IK90] KT97b] Domain decomposition methods for solving the state equation (1.1) for given q and f have been extensively studied. There is a vast literature of which we only mention some relevant classical papers [BPS86], BW86] and [MQ89] 2 The Domain Decomposition Approach Through out this work Omega is assumed to be a two dimensional, bounded, simply connected convex domain with piecewise smooth boundary. We decompose Omega into finitely many nonoverlapping subdomains. The decomposition is carried out in ....

Bramble J. H., Pasciak J. E., and Schatz A. H. (1986) An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp. 46: 361--369.

....[66] involves only one subregion solution per iteration. See Mandel and McCormick [65] for a comparison of FAC, the symmetric BEPS [4] preconditioner and the algorithm presented here and their theories. By considering the domain decomposition techniques presented by Bramble, Pasciak, and Schatz [6] that led to this algorithm, we can see that if the subregion problems (in Equation (51) and its sequels with updated guesses for P n b ) are solved exactly, then Q n 4 in Equations (53) and (54) is identically zero and the iterative method presented here requires only one subregion solution ....

J. H. Bramble, J. E. Pasciak, and A. H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp., 46 (1986), pp. 361--370.

....of the continuous problem. Establishing discrete extension theorems for some typical discretization schemes is an important, highly technical and delicate next step in developing the theory. For the di usion equation, discrete extension theorems are known in many cases, e.g. for the FEM, see [21, 52, 39, 45, 2, 40, 29, 48]. Discrete analogs of extension theorems of the present paper are apparently not yet obtained. One interesting class of applications of extension theorems has been developed in [7, 34, 8, 13, 9, 17, 51, 1, 10, 4, 18, 19, 35, 3, 6, 5, 20] where ecient preconditioned iterative methods for di erent ....

J. H. Bramble, J. E. Pasciak, and A. H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp., 46 (1986), pp. 361-369.

.... decomposition method is given by Douglas and the author in [11] Other domain decomposition methods (the Schwarz method, the Poincare Steklov operator method, and the Schur complement or substructuring method) can be found in, for example, Bjorstad and Widlund [1] Bramble, Pasciak and Schatz [4], De Roeck and Le Tallec [5] Le Tallec, De Roeck and Vidrascu [13] Lions [14] Mandel [17] Smith [22] and Widlund [23] The Schwarz methods involve cycling between overlapping subdomains whose unknown boundary data are updated by neighbors. The Poincare Steklov methods and the Schur complement ....

Bramble, J. H., Pasciak, J. E. and Schatz, A. H. 1986 An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp. 46 361-369.

....above equivalence to construct multilevel preconditioners for the linear system for the Lagrange multipliers. Then the mixed method solutions q h and u h are recovered via these multipliers. The construction of multilevel preconditioners for the mixed methods is inspired by the fundamental work [4], 16] where new systematic representations for preconditioners in the Neumann Dirichlet domain decomposition methods for conforming finite elements were suggested. The multilevel domain decomposition versions of these methods were outlined in detail in [17, 18] In addition, the superelement ....

J.H. Bramble, J.E. Pasciak, and A.H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp., 46 (1986), 361--369.

....on the substructures which form a non overlapping partition of the original domain. When a coarse solver is added, the rate of convergence can be made independent of the number of subregions. convergence. The method considered here has its roots in the early work by Bramble, Pasciak, and Schatz [6]; see also [20] That work is all for the H 1 case. There has been extensive work on the three dimensional case as well; see, e.g. Dryja, Smith, and Widlund [9] and the many references therein. We note that for all these iterative substructuring methods, the condition number of the relevant ....

James H. Bramble, Joseph E. Pasciak, and Alfred H. Schatz. An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp., 46(173):361-369, 1986.

....tioner in table 19, three dimensional tests with face preconditioners in table 21 and the combination of edge and face preconditioner in table 22. 5 BPS and Wire basket preconditioner for the Schur complement 5. 1 Definition There are lot of methods proposed in a series of papers starting with [BPS86] all referred to as BPS which might be a little confusing. The idea is to use sub domain problems for preconditioning the Schur system. W cross point Figure 9: Domain decomposition with a cross point. The preconditioners based on the eigen decomposition in the last chapter were not able to handle ....

J. H. Bramble, J. E. Pasciak, and A. H. Schatz. An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp., 46:361--369, 1986.

....with the matrix A. Since M is the matrix associated with a finite element discretization of the problem (23) it is well known that the spectral condition number of M Gamma1 A satisfies (M Gamma1 A) max (x;y;z) max (K(x; y; z; t) min (x;y;z) min (K(x; y; z; t) see e.g. [3] and references therein. Here, K is the matrix defined in (7) and max and min denote respectively the maximum and minimum eigenvalue of K at a specific time level. A FINITE ELEMENT METHOD FOR FULLY NONLINEAR WATER WAVES 9 For simplicity we consider this estimate closer in the 2D case, i.e. x; ....

J. H. Bramble, J. E. Pasciak and A. H. Schatz, An Iterative Method for Elliptic Problems on Regions Partitioned into Substructures, Math. of Comp., v. 46, 1986, pp. 361-369.

....i , and these are further triangulated into elements. H denotes the diameter of a typical substructure and h the diameter of one of its elements. We develop a domain decomposition algorithm similar to those considered by Bj rstad and Widlund [8] 9] Bramble, Pasciak, and Schatz [12] [13]; Dryja and Widlund [36] and Widlund [84] When using these methods, the variables interior to individual substructures are first eliminated. The resulting reduced system, the Schur complement, therefore only involves the variables associated with Gamma, the set of edges and vertices of the ....

....on Gamma ij . The same preconditioner was also discussed in Bj rstad and Widlund [9] and Bramble, Pasciak, and Schatz [12] Other preconditioners for these two subregion subproblems, such as the Neumann Dirichlet al..gorithm, were considered by Bj rstad and Widlund [9] Bramble, Pasciak, and Schatz [13]; Chan and Resasco [23] Chan and Keyes [22] Dihn, Glowinski, and P eriaux [30] and Golub and Mayers [40] A number of the resulting algorithms for the many substructure case are known to be almost optimal in the sense that the condition number is bounded by C(1 log(H=h) 2 : For a more ....

James H. Bramble, Joseph E. Pasciak, and Alfred H. Schatz. An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp., 46(173):361--369, 1986.

....with the matrix A. Since M is the matrix associated with a finite element discretization of the problem (23) it is well known that the spectral condition number of M Gamma1 A satisfies (M Gamma1 A) max (x;y;z) max (K(x; y; z; t) min (x;y;z) min (K(x; y; z; t) see e.g. [3] and references therein. Here, K is the matrix defined in (7) and max and min denote respectively the maximum and minimum eigenvalue of K at a specific time level. For simplicity we consider this estimate closer in the 2D case, i.e. x; z) 2 Omega 2 = 0; L] Theta [ GammaH; 0] The ....

J. H. Bramble, J. E. Pasciak and A. H. Schatz, An Iterative Method for Elliptic Problems on Regions Partitioned into Substructures, Math. of Comp., v. 46, 1986, pp. 361-369.

....H 1=2 ( Gamma) of Dryja [63] or the improved version [72] were based on the sine transform. The next step was to construct preconditioners for the case of many sub domains. Then, crosspoints are contained in the separator. Bramble, Pasciak and Schatz were able to treat this case with crosspoints [37]. This algorithm and its three dimensional generalization, the wirebasket preconditioner [38, 147] are based on a splitting of the space V Gamma = V coarse P k V edge k for d = 2, and V Gamma = V coarse P k V edge k P l V face l for d = 3, resp. 25) into one global coarse grid ....

J. H. Bramble, J. E. Pasciak, and A. H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp., 46 (1986), pp. 361--369.

....above equivalence to construct multilevel preconditioners for the linear system for the Lagrange multipliers. Then the mixed method solutions oe h and u h are recovered via these multipliers. The construction of multilevel preconditioners for the mixed methods is inspired by the fundamental work [7], 22] where new systematic representations for preconditioners in the Neumann Dirichlet domain decomposition methods for conforming finite elements were suggested. The multilevel domain decomposition versions of these methods were outlined in detail in [23] 24] and their multigrid versions ....

J.H. Bramble, J.E. Pasciak, and A.H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp., 46 (1986), 361--369.

....1 4 are typically satisfied for simple geometries as e.g. in our prototype problem depicted in Figure 1. Generally, conditions 1 3 are easily checked for a given geometry, whereas Condition 4 is harder to verify. This issue is carefully discussed in the paper by Bramble, Pasciak and Schatz [2], 3] The Ritz Galerkin discretization of (3.5) is defined as follows: Find p ffl;h 2 p dir V Omega ;h such that Z Omega r Delta [ ffl (rp ffl;h Gamma Q) dx = Z Omega 1 f dx Gamma Z Gamma neu g neu ds (5.1) for all 2 V Omega ;h . Similarly, the discrete approximation ....

J. H. Bramble, J. E. Pasciak and A. H. Schatz, An Iterative Method for Elliptic Problems on Regions Partitioned into Substructures, Math. of Comp., v. 46, 1986, pp.361-369

....balance the computational and memory requirement on a distributed memory parallel computer without sacrificing the communication overhead. 1 Introduction Domain decomposition is one of the most effective and popular technique for solving large scale numerical systems on parallel computers [5, 7]. This technique is used for finding solutions to partial differential equations by iteratively solving subproblems defined on smaller subdomains. Thus, it is a divide and conquer technique. When applying this technique, it is desirable to decompose the domain into subdomains with approximately ....

.... it is desirable to decompose the domain into subdomains with approximately the same computational work associated to them (for balancing the load) and to minimize communication between each subdomain and all other subdomains (for reducing total communication and communicational bottleneck) [5]. Part of this work was done while Daniel Spielman and Shang Hua Teng were visiting Universidad de Chile. y Dept. de Ingenier ia Matem atica, Fac. de Ciencias F isicas y Matem aticas, U. de Chile, Casilla 170 3, Correo 3, Santiago, Chile. mkiwi dim.uchile.cl, ....

J. H. Bramble, J. E. Pasciak, and A. H. Schatz. An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp., 46:361--9, 1986.

....V J 7 V J by (4.11) bv = k for all v 2 V J : It is easy to check that bs J v is equal to the trace of B vH and thus, K(bs J ) Let A i ( Delta; Delta) denote the form which results from (2.2) but with integration only over the subregion H . It is well known (cf. [3], 4] that A i (v H ; vH ) is uniformly (independent of H and J) equivalent to the square of the half Sobolev semi norm (jvj 1=2; Thus, s(v; v) is equivalent to 1=2; Gamma jvj 1=2; We now prove the following theorem. Theorem 4.2. The condition number K(bs J ) K(B ) is ....

J.H. Bramble, J.E. Pasciak and A.H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp. 46 (1986), 361--369.

....to the development of domain decomposition preconditioners. The first is the so called non overlapping approach and is characterized by the need to solve subproblems on disjoint subdomains. Early work was applicable to domains partitioned into subdomains without internal cross points [BW86] BPS86b] Dry89] To handle the case of cross points, Bramble, Pasciak and Schatz introduced in [BPS86a] algorithms involving a coarse grid problem and provided analytic techniques for estimating the conditioning of the domain decomposition boundary preconditioner, a central issue in the subject. ....

.... these techniques were extended to problems in three dimensions in [BPS89] and [Dry88] A critical ingredient in the three dimensional algorithms was a coarse grid problem involving the solution averages developed in [BPS87] Related work is contained in [CMW95] Nep91] Smi90] The papers [BPS86b] BPS86a] BPS87] BPS88] and [BPS89] developed domain decomposition preconditioners for the original discrete system. The alternative approach, to reduce to an iteration involving only the unknowns on the boundary, was taken in [BW86] BPX91] CMW95] and [Smi90] The difference in the two ....

Bramble J., Pasciak J., and Schatz A. (1986) An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp. 46: 361--369.

....to the development of domain decomposition preconditioners. The first is the so called non overlapping approach and is characterized by the need to solve subproblems on disjoint subdomains. Early work was applicable to domains partitioned into subdomains without internal cross points [1] [4], 14] To handle the case of cross points, Bramble, Pasciak and Schatz introduced in [5] algorithms involving a coarse grid problem and provided analytic techniques for estimating the conditioning of the domain decomposition boundary preconditioner, a central issue in the subject. Various ....

.... VASSILEV Subsequently, these techniques were extended to problems in three dimensions in [8] and [15] A critical ingredient in the three dimensional algorithms was a coarse grid problem involving the solution averages developed in [6] Related work is contained in [13] 20] 21] The papers [4], 5] 6] 7] and [8] developed domain decomposition preconditioners for the original discrete system. The alternative approach, to reduce to an iteration involving only the unknowns on the boundary, was taken in [1] 11] 13] and [21] The difference in the two techniques is important in ....

J.H. Bramble, J.E. Pasciak and A.H. Schatz , An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp. 46 (1986), 361--369.

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J H Bramble, J E Pasciak, and A H Shatz. An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp., 46:361--369, 1986.

No context found.

J. H. Bramble, J. E. Pasciak and A. H. Schatz, An Iterative Method for Elliptic Problems on Regions Partitioned into Substructures, Math. of Comp., v. 46, 1986, pp. 361-369.

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J. H. Bramble, J. E. Pasciak, and A. H. Schatz, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp., 46(1986), pp. 361-369.

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