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

.... nonconforming finite element or covolume methods have proven flexible and effective on incompressible fluid flow problems and biharmonic and plate problems ( 9, 8, tt] many researchers have been interested in studying multigrid methods for nonconforming finite elements or covolume methods ([6, 7, 2, 4, 1, 12]) In nonconforming multigrid methods, the intergrid transfer operators have important roles in conver gence. In this paper, we consider a covolume based intergrid transfer operator. This intergrid transfer operator needs less computation and neighborhood node information than previously ....

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

.... in the 1960s, and then by Astrakhantzev and Brandt in the early 1970s for the solution of these linear systems approximating PDEs (see Refs [3 8] In Brandt [7] it was claimed that these multigrid methods required only O (N) arithmetic operations; this was rigorously proved in Bank and Dupont [9] (also, see Refs [10 14] Actually, the multigrid methods are effective even in many cases where the classical iterative algorithms converge too slowly. The works of Refs [15 19] all describe parallel algorithms that take O(log N) arithmetic steps using N processors, and thus use the order of N ....

....L s, j = 1 . k, and for fixed c 0 and u 1. Assumption (3) holds for a wide class of PDEs (see, for instance, Refs [21, p. 29; 22] including the well posed linear PDEs, as well as many nonlinear PDEs. Such an assumption is routinely made in the analysis of the multigrid methods (e.g. Refs [6 9]) in particular, the auxiliary grid problems are said to be solved to the level of truncation defined by bound (3) see McCormick [20, p. 26] As a part of the weak pseudo regularity assumption, let us further assume that u(x) has been scaled so that lus(x) 1 for x eL s and for allj and ....

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

....the 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 ....

....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 ....

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

....which are not required to have full elliptic regularity. However, in all these three papers, only the W cycle algorithm with a sufficiently large number of smoothing steps was shown to converge using the standard proof of convergence of multigrid algorithms for conforming finite element methods [2]. We finally mention that the study of the NR Q 1 elements in the context of domain decomposition methods has been given in [13] In this paper we systematically study multigrid algorithms and multilevel preconditioners for discretizations of second order elliptic problems using the NR Q 1 ....

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

....(2.15) there are far fewer examples of operators B h known to satisfy (3.24) If the operator T gives rise to full elliptic regularity, then it is known that the W cycle multigrid algorithm with sufficiently many smoothings on each level gives rise to an operator B h which satisfies (3. 24) cf. [4]) Improved L ( Omega Gamma estimates depend upon elliptic regularity. We consider the adjoint boundary value problem in weak form: Given g 2 L find v 2 W such that (3.25) A(OE; v) OE; g) for all OE 2 W: Solutions of (3.25) exist and are unique since we have assumed uniqueness and ....

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

....method is presented in [23] These methods are based on Fourier analysis, hence are restricted to normal problems with constant coefficients. A more general (though less informative) approach is presented in [19] For certain finite element schemes, two level analysis is presented in [3] [4] [6] For automatic multigrid methods, however, no general two level analysis method have yet been developed. The aim of this work is to develop two level analysis methods and improved versions for automatic multigrid. The first part deals with possibly non separable problems. It is shown that ....

Bank, R. E.; and Dupont, T.: An Optimal Order Process for Solving Finite Element Equations. Math. Comp. 36 (1981), 35-51.

.... to efficiently 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 ....

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

....paper is to develop a technique for defining and analyzing multigrid algorithms for solving equations which involve discretizations of pseudo differential operators of negative order. Standard multilevel methods most often apply to discretizations of differential operators of positive order (cf. [1] [4] 7] 9] 16] 17] Let Omega be a polygonal domain in R . For nonnegative real s, let H ( Omega Gamma denote the Sobolev space of real valued functions with norm k Deltak s (see, 14] In addition, we shall use Sobolev spaces of negative index. For the purpose of this paper, we ....

....and last step in the above algorithm correspond to smoothing. The second step is the coarser grid correction step. More general versions of this algorithm involving increased smoothing on the various levels as well as more iterations in the coarser grid step are defined in the usual way (cf. [1], 3] 11] 15] Our theory extends to these algorithms as well (cf. 7] Typical presentations of the multigrid algorithm directly give rise to an iterative process with a linear reducer. This linear reducer is equal to I Gamma B j V j (cf. 4] Thus, the usual multigrid reduction process ....

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

....correction on nonconforming finite element spaces, while the second one has the coarse grid correction step established on conforming finite element spaces. The arguments in [6] and [7] which follow the standard proof of convergence of multigrid algorithms for conforming finite element methods [4], do not apply to the V cycle multigrid method. It is for this reason that we consider the second multigrid method here. We prove convergence of both the V cycle and W cycle algorithms for the second method in detail. While we have not found a proof for the convergence of the V cycle of the first ....

....(see Theorem 7.6 of [15] For our analysis, we also define k 2 U k such that (2.26) a k ( k ; v) f k ; v) 8v 2 U k : The usual error estimate for this finite element method is (2. 27) ku Gamma k k k Ch k kfk: If v 2 N k , we write v = c i OE i and define as in the standard case [4] jjjvjjj s;k = i s The Cauchy Schwarz inequality implies that ja k (w; v)j jjjwjjj 1 s;k jjjvjjj 1 Gammas;k for any s 2 IR and v; w 2 N k . Note that jjjvjjj 0;k = kvk and jjjvjjj 1;k = kvk k . We are in a position to prove that the kth level iteration NMG(k; g 0 ; f k ) when ....

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

....extension 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 ....

....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 expands the fine grid error in a product which reflects the effect of every coarser grid [6] This approach has been effectively applied even without the use of explicit regularity assumptions for the underlying differential equation. As far as we know, the ....

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

....the situation described above, where the special feature is the presence of the small parameter k. The framework is essentially that used in [3] where applications to elliptic problems are analyzed under weak assumptions, and our results and their proofs are close to those of earlier work, e.g. [1], 2] 4] and [7] We restrict our discussion here to the case of nested subspaces and inherited forms, and we make regularity assumptions that are satisfied for convex polygonal domains Omega Gamma This makes it possible to organize the theory in straightforward and compact manner, basing it on ....

....our results with those of [10] and [5] one may obtain estimates of the total error caused both by the discretization and the iterative solution of the algebraic equations. Various issues concerning multigrid methods for parabolic problems have been addressed in earlier work, for example, in [1], 8] 9] 12] 13] 14] 16] but in most cases (except [1] and [16] the convergence analysis is restricted to model problems with a uniform mesh, where Fourier methods can be applied. 2. Abstract multigrid analysis In the first subsection we define the multigrid algorithm and prove some ....

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

.... rate estimates but its application is limited to constant coefficient operators on only a few special domains [10] A more general approach based on the approximation properties of the spaces and the elliptic regularity properties of the underlying partial differential equation was pioneered in [1], 2] 11] More recently, an analysis for variational multigrid algorithms was provided which was not based on elliptic regularity [6] This analysis uses a multiplicative representation of the multigrid error propagator and has since been used and refined by other researchers [4] 13] 14] ....

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

....1=4 h e 0 h jjj 2 0 cjjj( h A h ) 1=4 (I h Gamma h A h ) m jjj 2 0 jjjA 1=4 h e 0 h jjj 2 0 ; respectively, jjje m h jjj 2 1 cjjj( h A h ) 1=4 (I h Gamma h A h ) m jjj 2 0 jjje 0 h jjj 2 1=2 . By standard arguments for positive definite operators (see, e.g. [1], 11] it is easy to see that jjj( h A h ) 1=4 (I h Gamma h A h ) m jjj 0 cm Gamma1=4 = c (m) 59) and therefore, by the help of Lemma 4, jjje m h jjj 1 c (m)jjje 0 h jjj 1=2 c (m)h Gamma1 ke 0 h k 0 : Analogously, for s = 3=2 , there holds A 3=4 h e m h = A 3=4 h ....

....ke m 1 h k 0 C (m)ke 0 h k 0 ; where (m) m Gamma1=4 . Additionally, if condition II is valid, we obtain ke m 1 h k 0 C (m) 2 ke 0 h k 0 : Hence, our method is convergent if the number m of smoothing steps is sufficiently large. Using standard inductive arguments (see also [1], 2] 17] we can verify the following Theorem. Theorem 2. Convergence of the k level scheme with p 2) Performing one step of MG(k;u 0 k ; g k ) with m damped Jacobi steps there holds with condition I and m sufficiently large ku k Gamma MG(k;u 0 k ; g k )k 0 C (m)ku k Gamma u 0 k k ....

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

....jjjujjj 1 Gammaff 2 jjjujjj ff 4 ; 0 ff 1: The norm equivalence jjjujjj s i juj s ; 0 s 2; is easy to see as follows. The cases s = 0 and s = 2 follow from the definition of the discrete norms. The result for 0 s 2 follows by interpolating the operators I and Q k ; cf. Bank and Dupont[1]. For polynomial or piecewise polynomial elements, the result can be extended to the case 2 s 5=2 based on the same reasoning as in [3] where it is shown that V k ae H 1 s and jjjujjj s i juj s with 0 s 3=2 for C 0 polynomial elements V k . We do not know however whether or not this ....

R. BANK AND T. DUPONT, An optimal order process for solving finite element equations, Math. Comp., 36 (1981), pp. 35--51.

.... Method for the P Version of the Finite Element Analysis Multigrid methods have been regarded as one of the most promising iterative methods for solving systems of linear equations arising from the discretization of partial differential equations by either the FDM or the h version of the FEM [6, 14, 15]. For example, it has been shown in [16] that for the Poisson equation in a rectangular domain the convergence factor of a multigrid V cycle method is bounded away from one independent of the mesh size h. However in general, there are two kinds of finite element methods: the h version and the ....

R. E. Bank and T. Dupont. "An Optimal Order Process for Solving Finite Element Equations". Math. Comp, 36(153):pp. 35--51, 1981.

....45] In all these earlier papers except in [26] only the W cycle multigrid methods have been shown to converge under the assumption that the number of smoothing iterations on all levels is sufficiently large. The methodology developed for the multigrid methods of conforming finite elements in [4] has been extensively employed to analyze the nonconforming multigrid methods; the convergence study is based on establishment of the so called smoothing and approximation properties and analysis of a two level scheme. Multigrid methods for nonconforming finite elements have the feature that the ....

....are noninherited. Consequently, the convergence proof of the conforming multigrid methods introduced in [6] does not apply to the nonconforming case since coarse to fine intergrid transfer operators for nonconforming finite elements do not preserve the energy norm. That is why the approach in [4] has been mainly exploited in the analysis of the nonconforming multigrid methods in the last decade. In multigrid methods for nested conforming finite elements the multilevel finite element spaces are nested and the quadratic forms are inherited. The purpose of this paper is to analyze intergrid ....

R. BANK AND T. DUPONT, An optimal order process for solving finite element equations, Math. Comp., 36 (1981), pp. 35--51.

.... z MG(k, z 0 , g) H 1(## # # z z 0 H 1(## . 3.3) Remark. Since the problem (3. 2) is symmetric positive definite and the V k s are conforming and nested, these assumptions are satisfied by either a V cycle or a W cycle multigrid algorithm with any number of smoothing steps (cf. [2], 6] 7] 8] Substituting (2.9) into (1.1) we obtain the following boundary value problem for w: #w = # # f J # j=1 # ##L # j #s # j,# # # J # j=1 # ##L j # j,# #s j,# in# , 3.4) w = 0 on ## . Note that #s j,# and #s # j,# # C # (# ) and that the functions s ....

....the definitions of a k and b k . By (4.6) and (4.21) we have w w k H 1 (## # w w k H 1 (## w k w k H 1 (## (4.25) # a k b k # # h m 1 # k #f# H m . OVERCOMING CORNER SINGULARITIES BY MULTIGRID 1891 Remark. It follows from a standard argument (cf. [2]) that cost of computing u k # # number of elements in T k . 4.26) Remark. If (m 1) # j #) is not an integer for 1 # j # J , then the function w belongs to H m 2 (## #H 1 0(## (cf. 18] and the # dependence in (2.16) 4.22) 4.24) and (4.26) disappears. Remark. When #(# j #) ....

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

....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 ....

....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 reduce to proving the so called strengthened Cauchy Schwarz inequality (see, [2, 3, 18, 28]) The case of different elements on the same mesh can often be analyzed by comparing them when mapped to a reference element. Comparisons between conforming and nonconforming elements are then straightforward. Results for mixed and conforming finite element pairs are given in [13] and for mixed ....

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

....correction on nonconforming finite element spaces, while the second one has the coarse grid correction step established on conforming finite element spaces. The arguments in [6] and [7] which follow the standard proof of convergence of multigrid algorithms for conforming finite element methods [4], do not apply to the V cycle multigrid method. It is for this reason that we consider the second multigrid method here. We prove convergence of both the V cycle and W cycle algorithms for the second method in detail. While we have not found a proof for the convergence of the V cycle of 4 ....

....analysis, we also define k 2 U k such that (2.26) a k ( k ; v) f k ; v) 8v 2 U k : The usual error estimate for this finite element method is (2. 27) ku Gamma k k k Ch k kfk: 12 ZHANGXIN CHEN If v 2 N k , we write v = P mk i=1 c i OE i and define as in the standard case [4] jjjvjjj s;k = mk X i=1 c 2 i s i 1=2 : The Cauchy Schwarz inequality implies that ja k (w; v)j jjjwjjj 1 s;k jjjvjjj 1 Gammas;k for any s 2 IR and v; w 2 N k . Note that jjjvjjj 0;k = kvk and jjjvjjj 1;k = kvk k . We are in a position to prove that the kth level iteration NMG(k; ....

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

....= F; I k k Gamma1 ) Gamma a k (g m ; I k k Gamma1 ) 8 2 N k Gamma1 : From (6.10) the multigrid approximate solution b oe k to oe k is defined in R 2 E k by (8. 6) b oe k = Gammaff Gamma1 k Phi rbz k fl R (f k Gamma d k PWk b z k )j RrfiR (x; y) Psi : The standard argument [2], 3] 4] for the convergence analysis of the multigrid algorithm (8.5) applies here if we prove that I k k Gamma1 is bounded and reduces to the natural injection on continuous bilinear functions. Although the second fact is false, it is true after a modification of the definition of I k ....

....kfk: 8.10) Proof. Equations (6.10) 8.6) and (8.7) imply equation (8.9) since fl R = O(h 2 k ) and krfi R k = O(h Gamma1 k ) Equation (8.10) follows with (7. 1) the bound is proportional to kfk because c = 0) It can be seen that the total work performed in obtaining b z k is O(n k ) [2]; thus, the cost to compute b oe k is also O(n k ) MIXED METHODS AS NONCONFORMING METHODS 21 Since b oe k belongs to b V k = fv : vj R = a 1 R a 2 R x; a 3 R a 4 R y) a i R 2 IR; 8R 2 E k g; but not necessarily to V k , following [4] we introduce the averaging operator k : ....

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R. Bank and T. Dupont, An optimal order process for solving finite element equations, Math. Comp. 36 (1981), 35--518.

....p (M k Delta; Delta) l 2 : By slight modification of the proof of the convergence theorem of CMG algorithm, we obtain the convergence theorem of the multigrid algorithm containing N k instead of M k with respect to (N k Delta; Delta) 1=2 l 2 which is equivalent to jjj Delta jjj 0;k . See [2] for more detail. For this specific experiment on the unit square we take N k = diag(M k ) as suggested in [2] which allows the use of an under relaxed Jacobi scheme of smoothing. Most rows of the = 10 = 100 = 1000 smoothing WU iter WU iter WU iter 1 2167 (1626) 7715 (5788) 10577 (7935) 2 ....

....we obtain the convergence theorem of the multigrid algorithm containing N k instead of M k with respect to (N k Delta; Delta) 1=2 l 2 which is equivalent to jjj Delta jjj 0;k . See [2] for more detail. For this specific experiment on the unit square we take N k = diag(M k ) as suggested in [2], which allows the use of an under relaxed Jacobi scheme of smoothing. Most rows of the = 10 = 100 = 1000 smoothing WU iter WU iter WU iter 1 2167 (1626) 7715 (5788) 10577 (7935) 2 2127 (798) 7052 (2645) 9272 (3478) 3 2079 (520) 6378 (1595) 8074 (2019) 4 2032 (381) 5823 (1092) 7161 (1343) ....

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

....a super linear speed up and a high efficiency on a large scale, MIMD multiprocessor computer. Experiments are presented for both the Intel Paragon and the IBM SP2. 1. Introduction. The multigrid method is an optimal order numerical scheme for solving a broad class of partial differential equations [4, 6, 12, 24]. With a parallel smoother it can be simply parallelized by using domain partitioning techniques, leading to a simple parallel multigrid method [7, 23] However, with a hierarchical structure, the simple parallel multigrid method suffers a deficiency of parallelism on coarser grids. In fact, the ....

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

....method is presented in [23] These methods are based on Fourier analysis, hence are restricted to normal problems with constant coefficients. A more general (though less informative) approach is presented in [19] For certain finite element schemes, two level analysis is presented in [3] [4] [6] For automatic multigrid methods, however, no general two level analysis method have yet been developed. The aim of this work is to develop two level analysis methods and improved versions for automatic multigrid. The first part deals with possibly non separable problems. It is shown that for ....

Bank, R. E.; and Dupont, T.: An Optimal Order Process for Solving Finite Element Equations. Math. Comp. 36 (1981), 35-51.

....[5, 6] The prototype of the multigrid convergence theory is that For some number of smoothing steps the multigrid process is a contraction for some norm. Moreover, the contraction number is independent of the mesh size h. This was proved for conforming multigrid methods by Bank and Dupont[1]. Braess and Hackbusch[2] and Hackbusch[8] proved this for the V cycle with one smoothing step. For the nonconforming multigrid method, this was proved by Braess and Verfurth[3] and Brenner[4] for the W cycle under the condition that each iteration step contains many smoothing steps. The method ....

....i ; j ) L 2 = ffi ij ( the Kronecker delta) such that a k ( i ; v) i ( i ; v) L 2 for all v 2 V k . From the inverse estimate (3) there exists C 0 such that i Ch Gamma2 k : 6) The same results hold for the conforming finite element spaces. The norm jjjvjjj s;k is defined (see [1]) as follows: jjjvjjj s;k : n k X i=1 s i 2 i 1=2 where v = n k X i=1 i i 2 V k : 7) Moreover, jjjvjjj 0;k = kvk L 2 and jjjvjjj 1;k = kvk k : 8) And, the Cauchy Schwarz inequality implies ja k (v; w)j jjjvjjj 1 t;k jjjwjjj 1 Gammat;k for any t 2 R and v; w 2 V k . ....

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

....operator. However, most of these papers focus on the finite element methods. Restricting to the variational form, Garlerkin form is a natural built in construction under these methods. With the regularity assumption of the partial differential equations, some convergence proofs are established, [2, 7, 8, 9]. The finite element method induces a nested set of smooth function subspaces. The Aubin Nitsch trick has been commonly used in these proofs. The multigrid method generated from finite difference method no longer induces a nested set of smooth subspaces where similar regularity estimate ....

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

....; where n is a positive integer independent of k and g k 2 V k is defined by (g k ; v) k = Z Omega 0 f N X j=1 X 2L j j; k Deltas j; 1 A v dx 8 v 2 V k : 3.35) We then define u k by u k = N X j=1 X 2L j j; k s j; w k : 3.36) Remark 3.2. A standard argument (cf. [5]) shows that the costs for both algorithms are proportional to dim V k . 10 S. C. BRENNER AND L. Y. SUNG 4 Convergence Analysis First we need to address the convergence of the standard k th level iteration. A key ingredient of which is the equivalence of Sobolev norms and certain ....

....It is clear that jjjvjjj 1 = jvj H 1( Omega Gamma 8 v 2 V k ; 4.2) jjjvjjj 0 kvk L 2( Omega Gamma 8 v 2 V k : 4.3) We want to establish the equivalence of jjj Delta jjj s and k Delta k H s( Omega Gamma on V k for 0 s 1 and s 6= 1=2. For a regular polygonal domain this is done (cf. [5], 12] by using the interpolation theory for Banach spaces. However, the assumption on Omega for the standard interpolation theory for Sobolev spaces (cf. 28] is not satisfied by a polygonal domain with cracks. In fact, even the definition of H s( Omega Gamma is more subtle for a ....

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

....#, independent of h k , such that 0 # 1 # (N k U, U) l 2 (M k U, U) l 2 # # #U # # k , U #= 0 . By slight modification of the proof of the convergence theorem of the CMG algorithm, we obtain the convergence theorem of the multigrid algorithm containing N 1 k instead of M 1 k . See [2] for more detail. While M 1 k S k and E k 1 k map # k to # # k and # # k 1 , respectively, N 1 k S k and N 1 k 1 (E k k 1 ) t M k do not preserve this property. Therefore, the multigrid solution obtained using N k does not belong to # # k . In this paper, we take N k as the diagonal of ....

R. E. BANK AND T. DUPONT, An optimal order process for solving finite element equations, Math. Comp., 36 (1981), pp. 35--51.

....paper is to develop a technique for defining and analyzing multigrid algorithms for solving equations which involve discretizations of pseudo differential operators of negative order. Standard multilevel methods most often apply to discretizations of differential operators of positive order (cf. [1] [4] 7] 9] 16] 17] Let Omega be a polygonal domain in R 2 . For nonnegative real s, let H s ( Omega Gamma denote the Sobolev space of real valued functions with norm k Deltak s (see, 14] In addition, we shall use Sobolev spaces of negative index. For the purpose of this paper, we ....

....and last step in the above algorithm correspond to smoothing. The second step is the coarser grid correction step. More general versions of this algorithm involving increased smoothing on the various levels as well as more iterations in the coarser grid step are defined in the usual way (cf. [1], 3] 11] 15] Our theory extends to these algorithms as well (cf. 7] Typical presentations of the multigrid algorithm directly give rise to an iterative process with a linear reducer. This linear reducer is equal to I Gamma B j V j (cf. 4] Thus, the usual multigrid reduction process ....

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

....15, 44] In all these earlier papers except in [25] only the W cycle multigrid methods have been shown to converge under the assumption that the number of smoothing iterations on all levels is sufficiently large. The methodology developed for the multigrid methods of conforming finite elements in [4] has been extensively employed to analyze the nonconforming multigrid methods; the convergence study is based on establishment 1991 Mathematics Subject Classification. Primary 65N30, 65N22, 65F10. Key words and phrases. Multigrid methods, nonconformingand mixed finite elements, second and ....

....are noninherited. Consequently, the convergence proof of the conforming multigrid methods introduced in [6] does not apply to the nonconforming case since coarse to fine intergrid transfer operators for nonconforming finite elements do not preserve the energy norm. That is why the approach in [4] has been mainly exploited in the analysis of the nonconforming multigrid methods in the last decade. In multigrid methods for nested conforming finite elements the multilevel finite element spaces are nested and the quadratic forms are inherited. The purpose of this paper is to analyze intergrid ....

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

....1=4 h e 0 h jjj 2 0 cjjj( h A h ) 1=4 (I h Gamma h A h ) m jjj 2 0 jjjA 1=4 h e 0 h jjj 2 0 ; respectively, jjje m h jjj 2 1 cjjj( h A h ) 1=4 (I h Gamma h A h ) m jjj 2 0 jjje 0 h jjj 2 1=2 . By standard arguments for positive definite operators (see, e.g. [1]) we get jjj( h A h ) 1=4 (I h Gamma h A h ) m jjj 0 cm Gamma1=4 ; 37) and, by Lemma 3, jjje m h jjj 1 cm Gamma1=4 jjje 0 h jjj 1=2 cm Gamma1=4 h Gamma1 ke 0 h k 0 : 3. The multigrid algorithm and its analysis 8 The main work was already done in the preceding ....

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

....q p k Gamma1 ) k (A k I k k Gamma1 q p k Gamma1 ; I k k Gamma1 q p k Gamma1 ) k : 4) 4. Postsmoothing step Analogously to the presmoothing step, apply m smoothing steps to u m 1 k , to obtain u 2m 1 k . The proof for the convergence of our multigrid algorithm combines as usual (see [1]) a smoothing property and an approximation property. Here we show the results for constant ff k = 1 and no postsmoothing, e i denotes the error u k Gamma u i k . Lemma 1. smoothing property) jje m jj k Ch Gamma1 k 1 4 p 4m 1 jjje 0 jjj 1;k : 5) Lemma 2. approximation property) ....

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

....analysis. Numerical comparisons with other existing methods are provided. AMS (MOS) subject classifications. 65N55, 65N30, 65F10 1. Introduction The multigrid method provides optimal order algorithms for solving large linear systems of finite element and finite difference equations (cf. [3]) The multigrid theory is well established (cf. 12, 16, 3, 15 and 5] However, in many situations, the multi level discrete spaces are nonnested due to the nature of the underlying finite elements ( 4, 7 8, 18, 25] or due to the special structures of grids (cf. 6, 920, 26, 27] Some special ....

....s k v; v) 8v 2 V k ; 0 s 2: We note that jvj 0;k = kvk L 2 and that jvj 1;k = p a k (v; v) Let k be the spectral radius of A k . By a simple calculation (cf. 10] on each triangle, we have the inverse estimate: k Ch Gamma2 k . We now define a multigrid scheme for solving (2. 2) cf. [3]) Definition 2.1(one kth level multigrid iteration) 1) For k = 1, the original problem (2.2) or the residual problem (2.8a) defined below is solved exactly. 2) For k 1, wm 1 will be generated from w 0 by the following two steps. We do m smoothings: 2.7) w l Gamma w l Gamma1 ; v) ....

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

.... the W cycle algorithms were proven to be convergent under the assumption that the number of smoothing iterations on all levels is big enough [8] 9] 12] The arguments in these earlier papers follow the standard proof of convergence of multigrid algorithms for conforming finite element methods [2], and do not apply to the V cycle algorithm. The second type of multigrid algorithm uses the nonconforming finite elements in the smoothing iterations on the finest level, but the P 1 conforming finite elements in the coarse grid corrections in the multilevel iteration. For this approach, uniform ....

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

....the 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 ....

....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 ....

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

....for matrices in normal form (i.e. A T A) in the case of convection diffusion (non symmetric) finite element elliptic equations. We prove, under H 2 regularity assumption (commonly imposed when studying convergence of multilevel methods in L 2 norm, cf. e.g. Bank and Dupont [BD81] and Goldstein, Manteuffel and Parter [GMP93] that in three space dimensions the AMLI method of hybrid type (see further Definition 3.2) is both of optimal order and spectrally equivalent to B = A T A. This method may be an alternative to the classical W cycle multigrid with sufficiently ....

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

....extension 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 ....

....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 expands the fine grid error in a product which reflects the effect of every coarser grid [6] This approach has been effectively applied even without the use of explicit regularity assumptions for the underlying differential equation. As far as we know, the ....

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

....(2.15) there are far fewer examples of operators B h known to satisfy (3.24) If the operator T gives rise to full elliptic regularity, then it is known that the W cycle multigrid algorithm with sufficiently many smoothings on each level gives rise to an operator B h which satisfies (3. 24) cf. [4]) Improved L 2 ( Omega Gamma estimates depend upon elliptic regularity. We consider the adjoint boundary value problem in weak form: Given g 2 L 2( Omega Gamma find v 2 W such that (3.25) A(OE; v) OE; g) for all OE 2 W: LEAST SQUARES APPROACH FOR FIRST ORDER SYSTEMS 15 Solutions of ....

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

.... to efficiently 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 ....

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

....which are not required to have full elliptic regularity. However, in all these three papers, only the W cycle algorithm with a sufficiently large number of smoothing steps was shown to converge using the standard proof of convergence of multigrid algorithms for conforming finite element methods [2]. We finally mention that the study of the NR Q 1 elements in the context of domain decomposition methods has been given in [13] In this paper we systematically study multigrid algorithms and multilevel preconditioners for discretizations of second order elliptic problems using the NR Q 1 ....

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

.... this operator, as mentioned in the introduction, it has been first shown in [6, 9] that the W cycle algorithm (i.e. 2) is convergent under the assumption that the number of smoothing iterations on all levels is big enough (following the standard proof of convergence for conforming methods [3, 4]) Then in [18] a convergence analysis for Algorithm 2.1 was given, which establishes optimal convergence properties of the W cycle multigrid algorithm and uniform condition number estimates for the variable V cycle preconditioner; see the theorem below. Since this operator does not preserve the ....

....V k . Also, since the operators in Examples 2 and 3 do not preserve the energy norm, we can only establish the optimal convergence properties of the W cycle multigrid algorithm and the uniform condition number estimates for the variable V cycle preconditioner via the standard convergence proof [3, 8], as in Example 1. With the three definitions above, we now state a convergence result, whose proof is given in [18] The convergence rate for Algorithm 2.1 on the kth level is measured by a convergence factor ffi k satisfying ja k ( I Gamma B k A k )v; v)j ffi k a k (v; v) 8 v 2 V k : ....

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

....multilevel scheme to solve discrete nonlinear systems of equations which arise from a standard weak formulation of the nonlinear partial differential equation. The framework of our analysis combines the multilevel iterative methods for linear finite element equations discussed in Bank and Dupont [3] and Bank [2] with the global approximate Newton setting of Bank and Rose [5, 4] Under appropriate conditions of elliptic regularity, we show that both the continuous and discrete solutions exist and that our scheme converges to an approximation within the discretization error of the continuous ....

....tolerance which is independent of the level. This implies that the total cost of solving a nonlinear problem of size N j is bounded by C Delta F (N j ) where F (N j ) is the cost of solving a linear problem of size N j and C 1. F (N j ) O(N j ) for the linear multigrid methods described in [3, 2]. Date: Received April 23, 1981; revised November 17, 1981 and January 29, 1982. 1980 Mathematics Subject Classification. Primary 65H10, 65F10, 65N20. This work was supported in part by the Office Of Naval research under grants N00014 80 C 0645 and N00014 76 C 0277. 1 2 Randolph E. Bank and ....

[Article contains additional citation context not shown here]

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

....extensions and present some examples of classes of spaces to which the method can be successfully applied. Our two level scheme can be generalized to a k level scheme for k 2. However, the rate of convergence which our analysis would predict depends on N if k does. We, as well as several others [2, 4, 12, 14], have obtained for various k level schemes convergence results comparable to our two level scheme. These multi level schemes are relatively complicated, and the requirements of the elliptic equation and the space M are more severe; e.g. the requirement that all the meshes are quasi uniform. When ....

....of this argument will work for any fixed number of levels, one would like the number of levels to depend on N . In this case the above analysis will fail to show that the rate of convergence is bounded less than one independent of h. However, such results have been obtained for multilevel schemes [2, 4, 5, 12, 14]. To do so, the concept of simple block iteration has been abandoned in favor of recursively defined algorithms. Furthermore, all presently known proofs explicitly or implicitly require some elliptic regularity, that the meshes T h j all be quasi uniform, 9 and that the spaces M h j satisfy ....

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

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R. E. Bank and T. F. Dupont, An optimal order process for solving finite element equations, Math. Comp. 36 (1981), no. 153, 35--51.

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R. E. Bank and T. F. Dupont. An optimal order process for solving finite element equations. Math. Comp., 36(153):35--51, 1981.

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

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

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

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R. Ba-ak and T. Dupont, An optimal order process for solving finite element equations, Mathematics of Computation 36, 35-51 (1981).

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