J. H. Bramble and J. E. Pasciak. Iterative techniques for time dependent Stokes problems. Computers Math. Applic., 33(1-2):13-30, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....This shows the importance of the inf sup condition (29) in the limiting case of steady flow for large Deltat the quotient (61) reduces to the quotient in (29) 31) and it follows that (56) is satisfied with S = fl in the steady state limit Deltat 1. Recent work by Bramble and Pasciak [1] has formally established that for finite Deltat, the quotient (61) is bounded both above and below by constants independent of h and Deltat, although careful consideration is required in the separate cases Deltat h and Deltat h . Our analysis in section 4 suggests that a practical ....

J. Bramble and J. Pasciak, Iterative techniques for time dependent Stokes problems. Math. Applic., 33:13--30, 1997.

....authors have considered the use of symmetric positive definite block diagonal preconditioners for the indefinite algebraic systems arising from certain saddle point problems, such as the mixed formulation of scalar second order elliptic equations and the Stokes equations. See Bramble and Pasciak [5], Klawonn [16] Rusten and Winther [19] 20] Silvester and Wathen [21] and Wathen and Silvester [24] In designing and analyzing their preconditioners, these authors have exploited the fact that for these problems the upper left hand corner of the coefficient matrix, A h , is positive definite, ....

....2n ) where r = Gamma 1) 1) This upper bound for the reduction factor is in fact the same as one would obtain after n=2 iterations of the conjugate gradient method applied to the normal equations associated with the preconditioned system (5. 1) However, as for example discussed in [5] and [19] the minimum residual method will usually perform much better than a normal equation approach, and it is well understood that this phenomenon can be explained from the lack of symmetry around the origin of the spectrum of B t;h A t;h . 17 However, the difference in the performance of ....

J. H. Bramble and J. E. Pasciak, Iterative techniques for time dependent Stokes problem, to appear in Comput. Math. Appl..

....(# ) 2 ( y n 1 h #)y n 1 h , v h ) p n h , # v h ) L 2 (# ) f u n h , v h ) # v h # V h , # y n h , q h ) L 2 (# ) 0 # q h # P h . Each time step requires the solution of a generalized Stokes problem for which e#cient methods are available, see, e.g. [6]. The discrete admissible set is C h = U h # C, and for the discrete objective function we choose J h (y h , u h ) #t 2 n T # n=1 #y n h y n dh # 2 L 2 (# ) 2 # 2 #u h u dh # 2 U h , where y dh # V n T h and u dh # U h approximate y d and u d , ....

J. H. Bramble, J. E. Pasciak, Iterative techniques for time dependent Stokes problems, Comput. Math. Appl. 33 (1997) 13--30, approximation theory and applications.

....techniques into the flow part of GCT 1. 4 (Milestone CD2.2) We shall implement the so called BP conjugate gradient method for symmetric and indefinite systems of algebraic equations [20] The BP conjugate gradient method has been shown to converge more rapidly than the minimal residual method [21]. This approach involves an algebraic reformulation of the original indefinite system. The reformulated system is symmetric, positive definite, and amenable to solution by the conjugate gradient method, with or without preconditioning. An alternative way of computing the solutions of mixed finite ....

J.H. Bramble and J.E. Pasciak, Iterative Techniques for Time Dependent Stokes Problems, Brookhaven National Laboratory Report BNL-49970 (1994)

....energy norm [30] Consequently, the quantity minimized by preconditioned MINRES is a natural choice which is quasioptimal with respect to the energy norm. Second, although we have 5 considered only steady problems here, the same point of view can be adapted to the evolutionary Stokes equations [3, 6]. In this case, the matrix A consists of a linear combination of a velocity mass matrix and a discrete Laplace operator. Good preconditioners for the Schur complement operator require an approximate Poisson solve on the pressure space. The same considerations hold for this problem as in the ....

J. H. Bramble and J. E. Pasciak. Iterative techniques for time dependent Stokes problems. In W. Habashi, editor, Solution Techniques for Large-Scale CFD Problems, pages 201-216. John Wiley, New York, 1995.

....A = fA ij g, i; j = 1; 2; 3, is a 3 by 3 block matrix corresponding to the velocity components in the three momentum equations in (2.2) Since the matrix M is indefinite, the system (3.4) has to be reformulated in order for the preconditioned conjugate gradient method (PCG) to be applicable. In [BP94] a block preconditioning technique are introduced and we show in [Eik96] that this technique is efficient also for the extrusion problem. After some algebra on the rows, the system (3.4) is reformulated in such a way that the new coefficient matrix M is positive definite, M ae X Y oe ....

....the Schur complement BA Gamma1 B T , K = N h h 2 I ; 3.7) where h is the spatial resolution and N h is the solution operator on the pressure grid for a finite element approximation to a Neumann problem, that is w = N h f satisfies (rw; rOE) f; OE) for test functions OE. The theory in [BP94] shows that this preconditioner gives rise to convergence rates which can be bounded independently of the mesh size h. As a preconditioner for A we use a block diagonal matrix A Gamma1 0 with three copies of a preconditioner for the submatrix A 11 on the diagonal, see [Eik96] We have mostly ....

Bramble J. and Pasciak J. (1994) Iterative techniques for time dependent Stokes problems. Math. Comp. .

....(1.1) have also been proposed in the modelling of macrosegregation formation in binary alloy solidification, cf. 13] Systems of the form (1. 1) may also arise from time discretizations of the Navier Stokes equation, where the parameter # corresponds to the square root of the time step, cf. [3]. However, the study of such time discretizations is not the motivation for the present paper. The purpose of the present paper is to discuss a finite element method for the model problem (1.1) with convergence properties that are uniform with respect to the perturbation parameter #. In 2 we ....

J.H. Bramble and J.E. Pasciak, Iterative techniques for time dependent Stokes problem, Comput. Math. Appl. 33 (1997), pp. 13--30.

....system (1.1) have also been proposed in the modelling of macrosegregation formation in binary alloy solidi cation, cf. 13] Systems of the form (1. 1) may also arise from time discretizations of the Navier Stokes equation, where the parameter corresponds to the square root of the time step, cf. [3]. However, the study of such time discretizations is not the motivation for the present paper. The purpose of the present paper is to discuss a nite element method for the model problem (1.1) with convergence properties that are uniform with respect to the perturbation parameter . In x2 we will ....

J.H. Bramble and J.E. Pasciak, Iterative techniques for time dependent Stokes problem, Comput. Math. Appl. 33 (1997), pp. 13-30.

....well conditioned. Nevertheless, the robust wavelet based preconditioners, discussed in Section 6, would cover here the full range of possible values of ff. Now, for small ff, the Schur complement tends more and more to a second order operator, so that preconditioning becomes necessary; see [29]. Again an asymptotically optimal preconditioner based on the above wavelet bases, namely Algorithm 1 (CB) in Section 6.2, is proposed in [67] The concrete examples considered there are based on dual pairs OE; OE where OE is chosen as a B spline; see Section 4.4. All basis functions and ....

J. H. Bramble, J. Pasciak, Iterative techniques for time dependent Stokes problems, Preprint, 1994.

....However, we can also use the projection method as a preconditioner for the coupled system of equation, as proposed by Turek [53] This means using some kind of Laplace operator as a preconditioner for the pressure Schur complement. This approach has been used in [17] An analysis can be found in [14]. 6.2.1 The pressure correction scheme We turn our attention to the nonstationary incompressible Navier Stokes equations: u t u Delta ru Gamma Deltau rp = f r Delta u = 0; supplemented with the same boundary conditions as in the stationary case and additionally an initial ....

Bramble, J., Pasciak, J.: Iterative techniques for time dependent Stokes problem, preprint (1994).

....This shows the importance of the inf sup condition (29) in the limiting case of steady flow for large Deltat the quotient (61) reduces to the quotient in (29) 31) and it follows that (56) is satisfied with S = fl 2 in the steady state limit Deltat 1. Recent work by Bramble and Pasciak [1] has formally established that for finite Deltat, the quotient (61) is bounded both above and below by constants independent of h and Deltat, although careful consideration is required in the separate cases Deltat h 2 and Deltat h 2 . Our analysis in section 4 suggests that a ....

J. Bramble and J. Pasciak, Iterative techniques for time dependent Stokes problems. Math. Applic., 33:13--30, 1997.

....is independent of the mesh parameter h. Our analysis is also valid for inexact preconditioning blocks. Detailed proofs of theorems discussed here may be found in [Krz97] Block preconditioners for saddle point problems were discussed by many authors before, see for example [D y87] BP88] BP90] [BP97], RW92] SW94] Elm96b] ES96] ESW97] Elm96a] Kla98b] Kla98a] KS97] The approach we propose gives an application programmer a great opportunity to reuse, in an efficient way, existing very powerful methods (or software) like the domain decomposition or the multigrid method for ....

Bramble J. and Pasciak J. (1997) Iterative techniques for time dependent Stokes problems. Comput. Math. Appl. 33(1-2): 13--30. Approximation theory and applications.

....the importance of the inf sup condition (2.21) in the limiting case of steady flow for large Deltat the quotient (6.8) reduces to the quotient in (2.21) 2.23) and it follows that (6.3) is satisfied with S = fl 2 in the steady state limit Deltat 1. Recent work by Bramble and Pasciak [1] has formally established that for finite Deltat, the quotient (6.8) is bounded both above and below by constants independent of h and Deltat, although careful consideration is required in the separate cases Deltat h 2 and Deltat h 2 . Our analysis in section 4 suggests that a ....

J. Bramble and J. Pasciak, Iterative techniques for time dependent Stokes problems. Math. Applic., 33:13--30, 1997.

....it is clear that A j is symmetric positive definite and that (1:7) amounts to solving the saddle point problem A j B j B j 0 u j p j = f 0 : 1. 12) The treatment of (1:12) is in one way or another tied to the Schur complement K j : B j A Gamma1 j B j (see e.g. [4]) In case (1:4) and not too small K j is well conditioned. However, when employing implicit time stepping schemes for the non stationary case (1:1) the condition number deteriorates with decreasing time steps. Building up on recent investigations in [4] we will point out in Section 4 that ....

....j : B j A Gamma1 j B j (see e.g. 4] In case (1:4) and not too small K j is well conditioned. However, when employing implicit time stepping schemes for the non stationary case (1:1) the condition number deteriorates with decreasing time steps. Building up on recent investigations in [4], we will point out in Section 4 that the multiscale bases not only give rise to stable discretizations in the sense of (1:8) but lead also to a convenient efficient preconditioner replacing the approach based on solving Neumann problems proposed in [4] In Section 5 we construct a class of ....

[Article contains additional citation context not shown here]

J.H. Bramble and J.E. Pasciak, Iterative techniques for time dependent Stokes problems, Preprint, 1994.

....for stable and stabilised mixed spaces with these convenient and popular choices. Since we wish to concentrate on the stability issue, we consider here the steady state Stokes problem as mentioned in section 2. For consideration of the additional issue arising with time dependent problems see [4]. 4.1. Statement of the problem. As in section 2, we can express the discrete Stokes problem as Ax : A B t B GammafiS u p = f 0 (4.1) where A is the discrete vector Laplacian, B is the discrete gradient so that its adjoint B t is the discrete negative divergence and S ....

J. Bramble and J. Pasciak, Iterative techniques for time dependent Stokes problems, in Solution Techniques for Large-Scale CFD Problems, W. Habashi, ed., John Wiley, 1995, pp. 201-- 216.

....authors have considered the use of symmetric positive definite block diagonal preconditioners for the indefinite algebraic systems arising from certain saddle point problems, such as the mixed formulation of scalar second order elliptic equations and the Stokes equations. See Bramble and Pasciak [5], Klawonn [16] Rusten and Winther [19] 20] Silvester and Wathen [21] and Wathen and Silvester [24] In designing and analyzing their preconditioners, these authors have exploited the fact that for these problems the upper left hand corner of the coefficient matrix, A h , is positive definite, ....

....2n ) where r 2 = Gamma 1) 1) This upper bound for the reduction factor is in fact the same as one would obtain after n=2 iterations of the conjugate gradient method applied to the normal equations associated with the preconditioned system (5. 1) However, as for example discussed in [5] and [19] the minimum residual method will usually perform much better than a normal equation approach, and it is well understood that this phenomenon can be explained from the lack of symmetry around the origin of the spectrum of B t;h A t;h . However, the difference in the performance of ....

J. H. Bramble and J. E. Pasciak, Iterative techniques for time dependent Stokes problem, to appear in Comput. Math. Appl..

....equations. Such systems typically are obtained when multiplier or mixed discretization techniques are employed. Examples of these include the discrete equations which result from approximation of elasticity problems, Stokes equations and sometimes linearizations of Navier Stokes equations [4], 14] 15] 16] In addition, these systems result from Lagrange multiplier [2] 3] 24] and mixed formulations of second order elliptic problems [8] 21] 24] We shall consider iterative solution of an abstract saddle point problem. Let H 1 and H 2 be finite dimensional Hilbert spaces ....

....by D(v; w) i=1 rv i Delta rw i dx: We next identify approximation subspaces of (H 0( Omega Gamma4 In order to avoid unnecessary complexity of the presentation only a two dimensional example will be considered. The discussion here is very closely related to the examples given in [4] and [5] where additional comments and other applications can be found. We partition Omega 1 Gamma1 Gamma1 1 Fig. 5.1. The square mesh used for H 2 ; the support (shaded) and values for a typical OE ij . into 2n Theta 2n square shaped elements, where n is a positive integer and define h ....

J. Bramble and J. Pasciak, Iterative techniques for time dependent Stokes problems. Brookhaven National Laboratory Report BNL-49970 (1994).

....solutions are mostly isolated, i.e. there exists a neighborhood of and f in which each solution is unique. We refer the reader to [9] and [17] for additional discussion. We next define our finite element approximation subspaces. The discussion here is very closely related to the examples given in [3] and [2] where additional comments and other applications can be found. We partition Omega into 2n Theta 2n squares, where n is a positive integer and define h = 1=2n. Let x i = ih and y j = jh for i; j = 1; 2n. Each of the squares is further partitioned into two triangles by its diagonal ....

J.H. Bramble and J.E. Pasciak. Iterative techniques for time dependent Stokes problems. Computers Math. Applic., pages 13--30, 1997.

....means that the bifurcation of solutions is rare and branches of solutions can be computed. We refer the reader to [11] and [20] for additional discussion of the subject. We next define our finite element approximation subspaces. The discussion here is very closely related to the examples given in [2] and [3] where additional comments and other applications can be found. We partition Omega into 2n Theta 2n square shaped elements, where n is a positive integer and define h = 1=2n. Let x i = ih and y j = jh for i; j = 1; 2n. Each of the square elements is further partitioned into two ....

J.H. Bramble and J.E. Pasciak. Iterative Techniques for Time Dependent Stokes Problems. Inter. Jour. Computers and Math. with Applic. (to appear).

....equations. Such systems typically are obtained when multiplier or mixed discretization techniques are employed. Examples of these include the discrete equations which result from approximation of elasticity problems, Stokes equations and sometimes linearizations of Navier Stokes equations [4], 14] 15] 16] In addition, these systems result from Lagrange multiplier [2] 3] 24] and mixed formulations of second order elliptic problems [8] 21] 24] We shall consider iterative solution of an abstract saddle point problem. Let H 1 and H 2 be finite dimensional Hilbert spaces ....

....rv i Delta rw i dx: We next identify approximation subspaces of (H 1 0( Omega Gamma7 d and L 2 0( Omega Gamma4 In order to avoid unnecessary complexity of the presentation only a two dimensional example will be considered. The discussion here is very closely related to the examples given in [4] and [5] where additional comments and other applications can be found. We partition Omega Analysis of the inexact Uzawa algorithm 17 1 Gamma1 Gamma1 1 Fig. 5.1. The square mesh used for H 2 ; the support (shaded) and values for a typical OE ij . into 2n Theta 2n square shaped elements, ....

J. Bramble and J. Pasciak, Iterative techniques for time dependent Stokes problems. Brookhaven National Laboratory Report BNL-49970 (1994).

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J. H. Bramble and J. E. Pasciak. Iterative techniques for time dependent Stokes problems. Computers Math. Applic., 33(1-2):13-30, 1997.

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J. Bramble and J. Pasciak, Iterative techniques for time dependent Stokes problems. Math. Applic., 33:13-30, 1997.

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