J. H. Bramble and J. E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp., 49 (1987), pp. 311--329.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....not too big. Therefore the method is optimal in the order of computation. The analysis is confirmed by our numerical computation. 2 Analysis We first analyze the the iterated penalty method. We will define a multigrid iterated penalty method. Then we will apply the general theory developed in [BP87], BPX91] and [SZ92] to obtain a convergence analysis of the multigrid iterated penalty method. Theorem 1. Convergence of the iterated penalty method) For u n defined in (3) the following error reduction relation holds: jjju j Gamma u n jjj 1;j C r jjju j Gamma u n Gamma1 jjj 1;j ....

....the initial guess for the multigrid iteration is u n Gamma1 and u Gamma1 = 0. w n is defined by w n = w n Gamma1 rdiv u n . By the assumptions (8 9) and (10) it is standard to verify the regularity and approximation assumption introduced by Bramble et al. see (3. 2) in [BP87], also [BPX91] and [SZ92] a r (u Gamma P j Gamma1 ; u) C 2 fi jjjA j ujjj 2 0;j j fi a r (u; u) 1 Gammafi 8u 2 U j with C 2 fi = Cr 1 ff=2 and fi = ff=2 (ff is defined in (9) Therefore, we can get the following theorem by the theory of [BP87] Theorem 2. The error ....

[Article contains additional citation context not shown here]

Bramble J. H. and Pasciak J. E. (1987) New convergence estimates for multigrid algorithms. Math. Comp. 49: 311--329.

....solution in V k by u k u k I k k Gamma1 u k Gamma1 3. Post smoothing: do 2 times u h u k Gamma C k (f k Gamma A k u k ) One of the numbers of smoothing steps 1 and 2 may be zero, giving an algorithm with only pre smoothing or only post smoothing. Both 1 and 2 may depend on k (Bramble and Pasciak, 1987) or they may be determined adaptively (Brandt, 1977) The special version of the algorithm when the coarse grid problem (6) is solved exactly is called the two grid algorithm. The error in the two grid algorithm is transformed according to e k M k e k ; 7) where M k is the two grid operator, ....

....the F cycle (Mandel and Parter, 1990) where C 1 and C 2 do not depend on k. General smoothers such as Gauss Seidel relaxation can be treated with ease in this framework, see Mandel et al. 1988) for the symmetric position definite case and Cao (1988) for the nonsymmetric and indefinite case. Bramble and Pasciak (1987) obtained independently estimates for the V cycle and the W cycle and the symmetric, positive definite case, which are asymptotically equivalent the above, and, in addition, proved a bound independent of h k on the convergence factor of the variable V cycle, which employs const: 2 m Gammak ....

J. H. Bramble and J. E. Pasciak (1987). New convergence estimates for multigrid algorithms, Math. Comp., 49, pp. 311--329.

....an interesting tool, particularly in finite element multigrid solvers based on adaptively refined meshes. In many cases this is a slower solution method than more standard adaptive mesh multigrid methods, but has the rather distinct advantage of being provably convergent. Bramble and Pasciak [7] developed a regularity free theory. This theory requires that the solvers used on each level commute with the differential operator. Since this eliminates the standard variant of Gauss Seidel, many in the multigrid field do not consider this to be an advantage. The 1990 s A significant portion ....

J. H. Bramble and J. E. Pasciak, "New convergence estimates for multigrid algorithms," Math. Comp., vol. 49, pp. 311--329, 1987.

....norm. The construction of smoothers for the nonconforming method fits into the general theory of [7] We can use point and line Jacobi and Gauss Seidel smoothing procedures to define R k [12] for example. The smoothing estimates are a consequence of the general smoothing theory developed in [6]. Hence, it suffices to consider the remaining two conditions, which are defined as follows: 2:6) a k (I k v; I k v) C a k Gamma1 (v; v) 8 v 2 V k Gamma1 ; and (2:7) ja k ( I Gamma I k P k Gamma1 ) v; v)j C ff kA k vk 2 k ff a k (v; v) 1 Gammaff ; 8 v 2 V k ; for k = 2; ....

J. Bramble and J. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp., 49 (1987), 311--329.

....employed. The pre and post smoothing operators which we employ corresponded to red black Gauss Seidel iterations, where each smoothing step consisted of sweeps, with each sweep consisting of one sub sweep with the red points followed by one sub sweep with the black points. A variable v cycle [7] approach to accelerating multilevel convergence was employed, so that the number of pre and post smoothing sweeps changes on each level; in our implementation, the number of pre and post smoothing sweeps at level k was given by = 2 J Gammak , so that one pre and post smoothing was ....

J. H. Bramble and J. E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), no. 180, 311--329.

....2( Omega Gamma norm. The construction of smoothers for the nonconforming method fits into the general theory of [5] We can use point Jacobi or Gauss Seidel smoothing procedures to define R k [8] for example. The smoothing estimates are a consequence of the general smoothing theory developed in [3]. Hence, it suffices to consider the remaining two conditions, which are defined as follows: 2:2) a k (I k v; I k v) C a k Gamma1 (v; v) 8 v 2 V k Gamma1 ; and (2:3) ja k ( I Gamma I k P k Gamma1 ) v; v)j C ff kA k vk 2 k ff a k (v; v) 1 Gammaff ; 8 v 2 V k ; for k = 1; ....

J. Bramble and J. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp., 49 (1987), 311--329.

.... fffl 2m [a k ( Gamma a k (K m k; K m k; 1 Gamma ffi 2 )C ff (1 Gamma ff)fl Gammaff= 1 Gammaff) ffi 2 ]a k (K m k; K m k; 47) We finish the argument by choosing fl such that the two coefficients above are equal and bounded above by ffi (see Theorem 3 in [10] for details) 2 Our next result is for the multigrid V cycle (p = 1) It has been shown in [11] that in this case BL is symmetric and positive definite operator on ML . Therefore the V cycle can be used as a preconditioner for AL . The next theorem indicates that, under a mild assumption on ....

J. H. Bramble and J. E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp., 49 (1987), pp. 311--329.

....National Science Foundation Grant No. DMS 9007185 and by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University. In conjunction, there has been intensive research into the theoretical understanding of the convergence properties of these methods (cf. 1] [3], 6] 11] 12] 14] 16] 17] These results provide a uniform convergence rate (with respect to the number of grid levels) for the V cycle algorithm in the case of full elliptic regularity and a quasi uniform mesh. It was shown in [5] using a new general multigrid theory that uniform ....

....regularity and a quasi uniform mesh. It was shown in [5] using a new general multigrid theory that uniform estimates hold for the V cycle algorithm, with only one smoothing per grid per iteration, even in the absence of full regularity and in the presence of mesh refinements. The results in [1] [3], 5] 6] 11] 12] 14] 16] 17] were applied to the finite element method with stiffness matrix computed exactly. In practice, the stiffness matrix is usually computed approximately using a suitable numerical quadrature. Furthermore, many standard finite difference schemes can be ....

J.H. Bramble and J.E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. (1987) 49 pp. 311--329 .

....smoothing iteration on all spaces except the first (with mesh size 1 2) on which we solve directly. The resulting multigrid iterative procedure is described in, for example, 27] The multigrid preconditioner results from applying one step of the iterative procedure with zero starting iterate, [5]. The V cycle uses one pre and post Gauss Seidel iteration sweep where the directions of the sweeps are reversed in the pre and post smoothing iterations. This results in a symmetric preconditioning operator B h which satisfies (5.1) 74(T h v; v) B h v; v) T h v; v) for all v 2 W h : The ....

J.H. Bramble and J.E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), 311--329.

.... solve the discrete equations which arise in the numerical approximation of partial differential equations (see the references in [11] 15] 18] In conjunction, there has been intensive research into the theoretical understanding of the convergence properties of these methods (cf. 2] 3] 5] [6], 8] 10] 15] 17] 18] In this paper, we present a new general theory based on two assumptions which are different from those made in earlier works. By using the new theory, we are able to derive some surprising uniform convergence bounds for a number of problems. The earlier theories ....

....suggested that the rates of convergence for these applications deteriorated as the number of multigrid levels increased. Previously, there were two general approaches for proving convergence of multigrid algorithms. The first was based on the so called regularity and approximations assumption [6]. The verification of this hypothesis used both the approximation properties of the discrete method as well as the regularity properties of the approximated partial differential equation. The theory of [6] 13] only provides a uniform convergence rate for the V cycle algorithm in the case of full ....

[Article contains additional citation context not shown here]

J.H. Bramble and J.E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), 311--329.

....no supporting theory was included. There are three basic theories for providing estimates for V cycle multigrid algorithms. The first is the so called regularity and approximation theory and provides estimates as long as elliptic regularity results are available for the underlying operator V [2] [3]. The second theory requires the weakest hypotheses but gives rise to estimates which depend on the number of levels in the multigrid algorithm [7] The third theory does not require elliptic regularity and often leads to uniform estimates on the rate of iterative convergence [4] We prove the ....

....that integral operators of positive order are considered in [19] Finally, the results of numerical experiments are given in Section 6. These results are in agreement with the theory of earlier sections. 2. The multigrid algorithm. In this section, we describe the multigrid algorithm following [3]. We assume that we are given a nested sequence of finite dimensional inner product spaces M 0 ae M 1 ae : ae M j ae H As earlier described, let V( Delta; Delta) be a symmetric positive definite bilinear form on M j satisfying (1.1) We shall develop multigrid algorithms for computing ....

[Article contains additional citation context not shown here]

J.H. Bramble and J.E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), 311--329.

....of the full regularity V cycle estimates provided by Braess and Hackbush in [2] 1. Introduction. In this paper, we provide some new convergence estimates for multigrid algorithms. In recent years, there have been many advances in the understanding of multigrid algorithms (e.g. see [1] [3], 5] 6] 9] 11] 12] 15] 17] 19] Two apparently different analytical approaches have been developed. Historically, the first used a two level error recurrence and proceeded to develop estimates for the multilevel case by repeated application (cf. 1] 12] The second approach ....

....and W cycle algorithms (with one smoothing iteration per level) applied to discretizations of (1.1) if the coefficients and the domain were such that (1.2) held with ff = 1. Estimates for V cycle algorithms with ff 1 and one smoothing per iteration using the first analysis were provided in [3], 11] but only with deterioration depending on the number of levels. The second type of theory does not necessarily depend on explicit regularity estimates. Earlier results of this type were given in [10] 16] and provided regularity free estimates for the two level case. Extensions to the ....

[Article contains additional citation context not shown here]

J.H. Bramble and J.E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), 311--329.

....parameter related to the compressibility of the medium and all estimates are independent of this parameter. Finally, the corresponding algebraic systems can be easily preconditioned by using preconditioners for standard second order problems, a task which is well understood (see, e.g. 2] 5] [9], 11] 12] 37] The outline of the remainder of the paper is as follows. In Section 2 we present the equations of linear elasticity and derive stability estimates which are used for the leastsquares formulation. In Section 3 we introduce the nite element method spaces of piecewise ....

....the cost of computing the action of B h applied to an arbitrary vector should be much less than that of applying T h . For our application, low cost preconditioners are known for which (4. 3) holds with C 0 and C 1 independent of the mesh size and hence the number of unknowns (see, e.g. 2] 5] [9], 11] 12] 37] The least squares method which we shall consider is based on the following bilinear form de ned for functions in V h h as well as for suciently smooth functions in H 1 D( hhfu; pg; fv; qgii (L h (u; p) B h L h (v; q) L(u; p) L(v; q) h (u; p) ....

J.H. Bramble and J.E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp., 49 (1987) 311-329.

....algebraic systems can be easily preconditioned. The preconditioner for the algebraic system corresponding to the finite element least squares approximation only requires the construction of preconditioners for standard second order problems, a task which is well understood (see, e.g. 3] 6] [11], 13] 14] 27] 44] This means that efficient iterative schemes can be developed to compute the discrete least squares approximation. The first computable H Gamma1 norm was used by R. Falk in [29] to treat weakly the incompressibility condition r Delta u = 0 for Stokes problems. This ....

....the cost of computing the action of B h applied to an arbitrary vector should be much less than that of applying T h . For our application, low cost preconditioners are known for which (4. 3) holds with C 0 and C 1 independent of the mesh size and hence the number of unknowns (see, e.g. 3] 6] [11], 13] 14] 27] 44] The least squares method which we shall consider is based on the form hh(u; p) v; q)ii 1 j (L h (u; p) B h L h (v; q) L(u; p) L(v; q) h oe (u; p) oe (v; q) h; Gamma N [oe (u; p) oe (v; q) h;I (r Delta u flp; r Delta v flq) ....

J.H. Bramble and J.E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp., 49 (1987) 311-329.

....Science, Brookhaven National Laboratory, Upton, NY 11973. Email: pasciak bnl.gov x Department of Mathematics, Texas A M University, College Station, TX 77834, Email: xzhang math.tamu.edu since any general nested or nonnested multigrid result implies the corresponding twolevel result (see, e.g. [3, 4, 5, 6, 8, 9, 17] and many others) The only reason for considering two level methods on their own is that it is possible to prove results which are stronger than those obtained in the general multilevel setting. Two level results are easily developed for elements of the same type and slightly refined meshes and ....

J. Bramble and J. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp., 49 (1987), pp. 311--329.

....no supporting theory was included. There are three basic theories for providing estimates for V cycle multigrid algorithms. The first is the so called regularity and approximation theory and provides estimates as long as elliptic regularity results are available for the underlying operator V [2] [3]. The second theory requires the weakest hypotheses but gives rise to estimates which depend on the number of levels in the multigrid algorithm [7] The third theory does not require elliptic regularity and often leads to uniform estimates on the rate of iterative convergence [4] We prove the ....

....that integral operators of positive order are considered in [19] Finally, the results of numerical experiments are given in Section 6. These results are in agreement with the theory of earlier sections. 2. The multigrid algorithm. In this section, we describe the multigrid algorithm following [3]. We assume that we are given a nested sequence of finite dimensional inner product spaces M 0 ae M 1 ae : ae M j ae H Gamma1=2( Omega Gamma : As earlier described, let V( Delta; Delta) be a symmetric positive definite bilinear form on M j satisfying (1.1) We shall develop multigrid ....

[Article contains additional citation context not shown here]

J.H. Bramble and J.E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), 311--329.

....National Science Foundation Grant No. DMS 9007185 and by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University. In conjunction, there has been intensive research into the theoretical understanding of the convergence properties of these methods (cf. 1] [3], 6] 11] 12] 14] 16] 17] These results provide a uniform convergence rate (with respect to the number of grid levels) for the V cycle algorithm in the case of full elliptic regularity and a quasi uniform mesh. It was shown in [5] using a new general multigrid theory that uniform ....

....regularity and a quasi uniform mesh. It was shown in [5] using a new general multigrid theory that uniform estimates hold for the V cycle algorithm, with only one smoothing per grid per iteration, even in the absence of full regularity and in the presence of mesh refinements. The results in [1] [3], 5] 6] 11] 12] 14] 16] 17] were applied to the finite element method with stiffness matrix computed exactly. In practice, the stiffness matrix is usually computed approximately using a suitable numerical quadrature. Furthermore, many standard finite difference schemes can be ....

J.H. Bramble and J.E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. (1987) 49 pp. 311--329 .

....Thus, the above assumption on the smoother is equivalent to assuming that R k is positive definite. Note that R k A k = I Gamma K k K k . To estimate the rate of convergence of iteration (2.1) with Mg k ( Delta; Delta) defined by Algorithm 2. 1, we first derive, as in Bramble and Pasciak [2], a two level recurrence relation for the error operator of the V cycle multigrid algorithm. Let u = A Gamma1 k g. Define e 0 = u Gamma u 0 , e 1=3 = u Gamma u 1=3 ; e 2=3 = u Gamma u 2=3 and e 1 = u Gamma u 1 . Then by the definition of the multigrid algorithm and the ....

.... Gamma P k Gamma1 )v) C Gammaff k A( I Gamma P k Gamma1 )v; v) for all v 2 M k : This type of regularity and approximation condition, however, does not hold for the anisotropic problem, and therefore, we cannot directly apply the theory of Braess and Hackbusch [1] and Bramble and Pasciak [2, 3]. We shall provide a modification of the theory of Braess and Hackbusch. We first consider symmetric smoothers. In this case, the condition that kK k k A 1 is equivalent to the condition that the spectrum oe(K k ) ae [0; 1) Lemma 2.1 Assume that R k is symmetric and that kK k k A j kI Gamma R ....

Bramble, J. H., and Pasciak, J. E. New convergence estimates for multigrid algorithms. Math. Comp. 49 (1987), 311--329.

....of the full regularity V cycle estimates provided by Braess and Hackbusch in [2] 1. Introduction. In this paper, we provide some new convergence estimates for multigrid algorithms. In recent years, there have been many advances in the understanding of multigrid algorithms (e.g. see [1] [3], 5] 6] 9] 11] 12] 15] 17] 19] Two apparently different analytical approaches have been developed. Historically, the first used a two level error recurrence and proceeded to develop estimates for the multilevel case by repeated application (cf. 1] 12] The second approach ....

....and W cycle algorithms (with one smoothing iteration per level) applied to discretizations of (1.1) if the coefficients and the domain were such that (1.2) held with ff = 1. Estimates for V cycle algorithms with ff 1 and one smoothing per iteration using the first analysis were provided in [3], 11] but only with deterioration depending on the number of levels. The second type of theory does not necessarily depend on explicit regularity estimates. Earlier results of this type were given in [10] 16] and provided regularity free estimates for the two level case. Extensions to the ....

[Article contains additional citation context not shown here]

J.H. Bramble and J.E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), 311--329.

No context found.

J.H. Bramble and J.E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), 311--329.

....smoothing iteration on all spaces except the first (with mesh size 1 2) on which we solve directly. The resulting multigrid iterative procedure is described in, for example, 27] The multigrid preconditioner results from applying one step of the iterative procedure with zero starting iterate, [5]. The V cycle uses one pre and post Gauss Seidel iteration sweep where the directions of the sweeps are reversed in the pre and post smoothing iterations. This results in a symmetric preconditioning operator B h which satisfies (5.1) 74(T h v; v) B h v; v) T h v; v) for all v 2 W h : The ....

J.H. Bramble and J.E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), 311--329.

.... solve the discrete equations which arise in the numerical approximation of partial differential equations (see the references in [13] 18] 21] In conjunction, there has been intensive research into the theoretical understanding of the convergence properties of these methods (cf. 2] 3] 6] [7], 9] 12] 18] 20] 21] In this paper, we present a new general theory based on two assumptions which are different from those made in earlier works. By using the new theory, we are able to derive some surprising uniform convergence bounds for a number of problems. The earlier theories ....

....suggested that the rates of convergence for these applications deteriorated as the number of multigrid levels increased. Previously, there were two general approaches for proving convergence of multigrid algorithms. The first was based on the so called regularity and approximations assumption [7]. The verification of this hypothesis used both the approximation properties of the discrete method as well as the regularity properties of the approximated partial differential equation. The theory of [7] 16] only provides a uniform convergence rate for the V cycle algorithm in the case of full ....

[Article contains additional citation context not shown here]

J.H. Bramble and J.E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), 311--329.

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J. H. Bramble and J. E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp., 49 (1987), pp. 311--329.

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J. H. Bramble and J. E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), no. 180, 311--329.

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J. H. Bramble and J. E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp., 49 (1987), pp. 311--329.

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