R. E. Bank and T. Dupont. An optimal order process for solving elliptic finite element equations. Math. Comp., 36:35--51, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1) grid algorithm for solving the system (64) is independent of the discretization parameter h l Gamma1 , then we get a h l independent convergence rate j of the Algorithm 1. 17 Remark 3.1. The strengthened Cauchy inequality (76) for various bilinear forms a( was analysed by many authors [1, 2, 4, 8, 11, 12, 19, 21]. Maitre and Musy [12] calculated the constant fl for bilinear forms corresponding to scalar partial differential equations of second order. Jung [8] and Jung Langer Semmler [11] studied the dependence of fl on the Poisson ratio for linear elasticity problems in two and three dimension. Remark ....

R. E. Bank and T. F. Dupont, An optimal order process for solving elliptic finite element equations, Mathematics of Computations, 36 (1981), pp. 35--51.

....level 1 and work their way to some level j, using each level k, k j, both to generate an initial guess for level k 1 and for solving residual correction problems. Analysis of nested iteration algorithms in the context of this paper can be found in [12] more traditional analyses can be found in [2], 7] 8] and [17] In this paper, only correction schemes are considered. Define a k level (stan dard) correction multigrid scheme by Algorithm MG(k; f g ; x k ; f k ) 1) If k = 1, then solve A 1 x 1 = f 1 exactly or by smoothing (2) If k 1, then repeat i = 1; Delta Delta Delta ....

R. E. Bank and T. Dupont. An optimal order process for solving elliptic finite element equations. Math. Comp., 36:35--51, 1981.

....in the variational formulation of the b.v.p. The splitting of the finite element subspace V h into VH and T h is the basis of several iterative solvers. Examples for such solvers are conjugate gradient (cg) methods with two level h or p hierarchical preconditioners proposed by Bank and Dupont [8], Axelsson and Gustafsson [2, 4] see also [16] the cg method with algebraic multilevel preconditioners developed by Axelsson and Vassilevski [5, 6] or multigrid methods of the projection type as described by Meis and Branca [22] Braess [9, 10] Verf urth [25] Jung [14, 15] Schieweck [23] ....

....and T h , respectively, are to be solved. For the subproblem corresponding to VH usually a recursive multilevel strategy is used. The stiffness matrix resulting from the discretization of the subproblem on T h has a condition 13 number which is independent of the discretization parameter [2, 8, 15], but it is increasing with an increasing Poisson s ratio . Numerically determined estimates of the condition number ( in the case of triangulations with right isosceles triangles and in the case of a tetrahedron as shown in Figure 8 are plotted in Figure 9 and in Figure 10, respectively. A good ....

R. E. Bank and T. F. Dupont. An optimal order process for solving elliptic finite element equations. Math. Comput., 36:35--51, 1981.

....in the sense of (4) with parameter fl = 1=2: Lemma 1.1. A smoother in the sense of (4) fulfills kS j;m j v j k a c h Gammaff j m fffl j kv j k H 1 Gammaff 8v j 2 X j : Proof. This can be shown be an usual interpolation argument using discrete Sobolev norms like those introduced in [1] and their equivalence to the fractional Sobolev norms. We are now able to state and prove the main convergence estimate for the cascadic iteration (1) Theorem 1.2. The error of the cascadic iteration with a smoother as basic iteration can be estimated by ku Gamma u k a c X j=1 1 ....

R. E. Bank and T. F. Dupont. An optimal order process for solving elliptic finite element equations. Math. Comp. 36, 967--975 (1981).

....refinement of a given coarse mesh, algebraic multigrid strives to build artificial coarse spaces and the associated operators from a matrix associated with a given fine mesh. For further general information, see, e.g. 3, 10, 13, 15, 17, 27] First generation multigrid convergence theories [1, 2, 9] were based mainly on the concept of approximation properties between the grids and elliptic regularity. Early in multigrid history, it was recognized that a class of multigrid algorithms can be seen as a successive minimization of the energy in a set of directions consisting of coordinate vectors ....

R. E. Bank and T. Dupont, An optimal order process for solving elliptic finite element equations, Math. Comp., 36 (1981), pp. 35--51.

....of multigrid cycles is needed to obtain a solution with error of the same order as the discretization error, giving a method with optimal asymptotical computational complexity for the solution of the discrete equations. For more details, see, for example, Brandt (1977) Hackbusch (1981a, 1985a) Bank and Dupont (1981) or Mandel et al. 1987) II.B Finite Differences and Fourier Analysis From the start, the multigrid algorithm was motivated by modal analysis for constant coefficient problems on a uniform grid (Fedorenko, 1961, Brandt, 1977, Stuben and Trottenberg, 1982) In modal analysis, the problem is ....

....of V k with the imbedding (22) but this is not useful unless (24) holds. Again, for simplicity, let Am be symmetric and positive definite. Then (24) implies that the energy norm jjju k jjj = q u T k A k u k (25) coincides in all spaces V k . Note that jjjI k k Gamma1 jjj = 1: 26) Bank and Dupont (1981) studied multigrid convergence in this framework, obtaining mesh independent convergence factors for the two grid algorithm and the W cycle with sufficiently many smoothing steps. Because all usual iterative methods decrease the energy norm of the error and so does the coarse grid correction step, ....

[Article contains additional citation context not shown here]

R. E. Bank and T. Dupont (1981). An optimal order process for solving elliptic finite element equations, Math. Comp., 36, pp. 35--51.

....00 (F ) # #( #F,h ) 1 2 v, v# 0,F # v # V h 0 (F ) 4. 24) where #F,h is the discrete Laplace operator in V h 0 (F ) The above relation is equivalent to the following: #( #) 1 4 v# 0,F = # #( #F,h ) 1 4 v# 0,F # v # V h 0 (F ) which is well known; see Bank and Dupont [2], Xu [79, 82] 4.2.4. Discrete Sobolev inequalities. Let us first state the following inverse inequalities that hold for all v # V h(##6 #v# 1,#,# # h 1 #v# 0,#,# , 4.25) #v# 0,#,# # h n 2 #v# 0,# , 4.26) #v# 1,# # h 1 #v# 0,# , 4.27) #v# 1 2,## # h ....

R. Bank and T. Dupont, An optimal order process for solving elliptic finite element equations, Math. Comput., 36 (1981), pp. 35--51.

.... in the works of Southwell [92] and Wachpress [100] The modern idea was introduced and analyzed by Brakhage [15] and Fedorenko [49, 50] in the 1960 s, followed by Bachvalov [5] Multigrid methods have not been paid much attention in the 1970 s until the works of Astrachancer [2] Bank and Dupont [6], Brandt [19] Hackbusch [57] Nicolaides [83] and others showed that multigrid is indeed a very useful technique practically and theoretically. An enormous amount of progress has been achieved since then. Various multigrid methods have been developed, ranging from geometry specific to purely ....

R. E. Bank and T. Dupont. An optimal order process for solving elliptic finite element equations. Math. Comp., 36:35--51, 1981.

....in the variational formulation of the b.v.p. The splitting of the finite element subspace V h into VH and T h is the basis of several iterative solvers. Examples for such solvers are conjugate gradient (cg) methods with two level h or p hierarchical preconditioners proposed by Bank and Dupont [8], Axelsson and Gustafsson [2, 4] see also [16] the cg method with algebraic multilevel preconditioners developed by Axelsson and Vassilevski [5, 6] or multigrid methods of the projection type as described by Meis and Branca [22] Braess [9, 10] Verf urth [25] Jung [14, 15] Schieweck [23] ....

....and T l h , respectively, are to be solved. For the subproblem corresponding to VH usually a recursive multilevel strategy is used. The stiffness matrix resulting from the discretization of the subproblem on T l h has a condition number which is independent of the discretization parameter [2, 8, 15], but it is increasing with an increasing Poisson s ratio . Numerically determined estimates of the condition number ( in the case of triangulations with right isosceles triangles and in the case of a tetrahedron as shown in Figure 8 are plotted in Figure 9 and in Figure 10, respectively. A good ....

R. E. Bank and T. F. Dupont. An optimal order process for solving elliptic finite element equations. Math. Comput., 36:35--51, 1981.

....level 1 and work their way to some level j, using each level k, k j, both to generate an initial guess for level k 1 and for solving residual correction problems. Analysis of nested iteration algorithms in the context of this paper can be found in [12] more traditional analyses can be found in [2], 7] 8] and [17] In this paper, only correction schemes are considered. Define a k level (stan dard) correction multigrid scheme by Algorithm MG(k; f g k =1 ; x k ; f k ) 1) If k = 1, then solve A 1 x 1 = f 1 exactly or by smoothing (2) If k 1, then repeat i = 1; Delta Delta ....

R. E. Bank and T. Dupont. An optimal order process for solving elliptic finite element equations. Math. Comp., 36:35--51, 1981.

....to the most effective sequential methods for solving problems in a wide range of applications. For example in the treatment of partial differential equations they allow for a multilevel solver like multigrid or the BPX preconditioner, see [5, 10] as well as adaptive refinement techniques, see [1, 3]. Here, the aim is to obtain an approximation to the continuous solution within a prescribed error tolerance with an amount of work which is proportional to N , i.e. the number of unknown of the finest adapted grid. The actual approximation error should be smaller than some given . To this end, ....

R. E. Bank and T. Dupont, An optimal order process for solving elliptic finite element equations, Math. Comp., 36 (1981), pp. 35--51.

....the corresponding coefficient values. We now have to find a data structure which allows to store, to retrieve and to access these data efficiently. A first approach might be a binary tree structure. Tree data structures are quite common in many adaptive codes for the multilevel solution of PDEs [4, 5, 32, 36]. There, different trees represent the hierarchies of nodes, edges and elements, while entities on one level of a tree represent one grid. Refining the finest grid means adding new leaves to the tree. However, in order to administrate the nodes (unknowns) edges (stiffness matrix) and elements ....

....where a finer grid is needed to resolve the solution. We start with a very coarse grid and iterate the procedure, always adding new nodes. If a final error tolerance is matched, we have a solution on a fine adapted grid. The technique is the same as for other adaptive refinement methods [4, 5, 32, 36] and can be applied straightforwardly also in our case. The difference is, however, in the refinement process. We work node oriented not element oriented. If a node index (l; i) has been flagged for refinement then the nodes of then next level , which lie in its influence cone, i.e. the set of ....

R. E. Bank and T. F. Dupont. An optimal order process for solving elliptic finite element equations. Math. Comp., 36, 967-975, 1981.

....solver has optimal O(n) complexity for n unknowns, the same complexity is desirable for its implementation on refined grids and for the grid refinement procedure itself. One way to construct such an optimal order algorithm is to use tree data structures. This is described in more detail in [2,12,15]. Different trees represent the hierarchies of nodes, edges and elements, see also Figure 2. The components of the element tree, the node tree and the edge tree from the root down to a specific level collectively represent the grid on that level. A refinement step for the actual finest grid adds a ....

R. E. Bank and T. F. Dupont. An optimal order process for solving elliptic finite element equations. Math. Comp., 36:967--975, 1981.

....successive refinement of a given coarse mesh, algebraic multigrid strives to build artificial coarse spaces and the associated operators from a matrix associated with a given fine mesh. For further general information, see, e.g. 3,11,17, 22,24,34] First generation multigrid convergence theories [1,2,10] were based mainly on the concept of approximation properties between the grids and 1 This research was supported in part by NSF grant ECS 9725504 and the Czech academic grant VS 97156. 2 elliptic regularity. Early in multigrid history, it was recognized that a class of multigrid algorithms ....

R. E. Bank and T. Dupont, An optimal order process for solving elliptic finite element equations, Math. Comp., 36 (1981), pp. 35--51.

....developed the multigrid method in [86, 87] and applied it to variable coefficient, general, second order, nine point stencil discretizations on arbitrary domains. The analysis of the multigrid method was continued for finite difference stencils in [162] and for more general error norms in [8] leading to what is For the solution of integral equations Brackhage [30] proposed a two grid method even earlier. now called the classical multigrid theory: An abstract convergence proof based on an approximation and a smoothing property was established in [88] The first V cycle ....

R. E. Bank and T. Dupont, An optimal order process for solving elliptic finite element equations, Math. Comp., 36 (1981), pp. 35--51.

....coarse grid approximation P l v l 1 of a fine grid function u l in terms of the energy (u l ) T A l u l and by the fact that the constant function has zero energy because of (3. 2) This requirement is present in two level theories based on strengthened Cauchy inequality and its variants [3, 4, 6] and a similar requirement can be proved to be necessary for some methods [12] Since the basis of the finest space is assumed to decompose a constant, cf. 3.5) we only need to require that the columns of each prolongation matrix form a decomposition of unity n l 1 X j=1 P ij = 1; l = 1; ....

R. E. Bank and T. Dupont, An optimal order process for solving elliptic finite element equations, Math. Comp., 36 (1981), pp. 35--51.

....F;h ) 1=2 v; vi 0;F 8v 2 V h 0 (F ) 4. 24) where Delta F;h is the discrete Laplace operator in V h 0 (F ) The above relation is equivalent to the following k( Gamma Delta) 1=4 vk 0;F = k( Gamma Delta F;h ) 1=4 vk 0;F 8v 2 V h 0 (F ) which is well known, see Bank and Dupont [1], Xu [70, 72] 4.2.4. Discrete Sobolev inequalities. Let us first state that the following inverse inequalities hold for all v 2 V h( Omega Gamma7 kvk 1;1; Omega h Gamma1 kvk 0;1; Omega ; 4.25) kvk 0;1; Omega h Gamman=2 kvk 0; Omega ; 4.26) kvk 1; Omega h ....

R. Bank and T. Dupont. An optimal order process for solving elliptic finite element equations. Math. Comp., 36:35--51, 1981.

....18, 19, 21, 22] The use of zero energy modes has become a recognized way to capture the essential information needed to build an efficient iterative method. Since the first attempts to analyze AMG type methods, it was clear that the classical multigrid theory, which relies on elliptic regularity [1, 12, 20] will not apply, because this theory requires the use of properties of the underlying finite element spaces on all levels. The approach based on a strengthened Cauchy inequality [1, 2] or, equivalently, on the weak approximation property [5, 13, 14, 17] needs only assumptions that can be verified ....

....AMG type methods, it was clear that the classical multigrid theory, which relies on elliptic regularity [1, 12, 20] will not apply, because this theory requires the use of properties of the underlying finite element spaces on all levels. The approach based on a strengthened Cauchy inequality [1, 2], or, equivalently, on the weak approximation property [5, 13, 14, 17] needs only assumptions that can be verified computationally, but gave originally convergence estimates for two level methods only, and simple recursive estimates result in a convergence bound that approaches 1 as a geometrical ....

R. E. Bank and T. Dupont, An optimal order process for solving elliptic finite element equations, Math. Comp., 36 (1981), pp. 35--51.

....1) grid algorithm for solving the system (64) is independent of the discretization parameter h l Gamma1 , then we get a h l independent convergence rate j of the Algorithm 1. Remark 3.1. The strengthened Cauchy inequality (76) for various bilinear forms a( was analysed by many authors [1, 2, 4, 8, 11, 12, 19, 21]. Maitre and Musy [12] calculated the constant fl for bilinear forms corresponding to scalar partial differential equations of second order. Jung [8] and Jung Langer Semmler [11] studied the dependence of fl on the Poisson ratio for linear elasticity problems in two and three dimension. Remark ....

R. E. Bank and T. F. Dupont, An optimal order process for solving elliptic finite element equations, Mathematics of Computations, 36 (1981), pp. 35--51.

.... Gamma 1) grid algorithm for solving the system (64) is independent of the discretization parameter h l , then we get a h l independent convergence rate j of the Algorithm 1. Remark 3.1. The strengthened Cauchy inequality (76) for various bilinear forms a( was analysed by many authors [1, 2, 3, 8, 11, 13, 20, 22]. Maitre and Musy [13] calculated the constant fl for bilinear forms corresponding to scalar partial differential equations of second order. Jung [8] and Jung Langer Semmler [11] studied the dependence of fl on the Poisson ratio for linear elasticity problems in twoand three dimension. Remark ....

R. E. Bank and T. F. Dupont, An optimal order process for solving elliptic finite element equations, Mathematics of Computations, 36 (1981), pp. 35--51.

....solver has optimal O(n) complexity for n unknowns, the same complexity is desirable for its implementation on refined grids and for the grid refinement procedure itself. One way to construct such an optimal order algorithm is to use tree data structures. This is described in more detail in [1,12,9]. Different trees represent the hierarchies of nodes, edges and elements. All entities of a tree down to one level altogether represent one grid. Refining the finest grid means adding new leaves to the tree. Moreover, to run multigrid is actually to traverse the tree several times and to perform ....

R. E. Bank and T. F. Dupont. An optimal order process for solving elliptic finite element equations. Math. Comp., 36:967--975, 1981.

....procedures. 1. Introduction. In this paper, we prove the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations. While many convergence proofs already exist (e.g. Astrakhantsev [2] Bakhvalov [4] Bank and Dupont [5], Braess and Hackbusch [7] Douglas [10, 11] Federenko [12] Hackbusch [15, 17, 14, 16] Maitre and Musy [19] Nicolaides [20] Van Rosendale [21] Verfurth [23] Wesseling [24] and Yserentant [25] our assumptions and proof techniques are different and (we believe) enlightening. While we are ....

....in which the sequence of discrete problems (2.1) associated with M 1 ; M 2 ; is solved. The solution of the j th problem serves as the initial guess for the (j 1) st. Since analysis of several such schemes for both linear and nonlinear problems is available elsewhere (e.g. Bank and Dupont [5], Bank and Rose [6] Douglas [10, 11] Hackbusch [15, 14, 16] we do not repeat that analysis here. Instead, we obtain reasonably sharp estimates for the spectral radius of the iteration matrix associated with Algorithm MG. There are two main components of Algorithm MG when j 1: smoothing and ....

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R. E. Bank and T. Dupont, An optimal order process for solving elliptic finite element equations, Math. Comp., 36 (1981), pp. 35--51.

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R. E. Bank and T. Dupont. An optimal order process for solving elliptic finite element equations. Math. Comp., 36:35--51, 1981.

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R. E. Bank and T. F. Dupont. An optimal order process for solving elliptic finite element equations. Math. Comp., 36:35--51, 1981.

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