Z. Chen, R. E. Ewing, and R. Lazarov. Domain decomposition algorithms for mixed methods for second order elliptic problems. Math. Comp., 65:467-490, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....be accomplished by using the Schur complement as in algorithms of Uzawa type, via the introduction of additional Lagrange multipliers and elimination of the vector variable, or via the use of divergence free bases and elimination of the scalar variable. Examples of such approaches can be found in [4, 6, 8, 9, 14, 15, 16, 17, 20, 22, 24, 32]. Finally, in this section we shall consider an application of the preconditioning of the k dependent operator associated to the bilinear form (6.1) Consider the system obtained by applying the mixed finite element method to the singular perturbation Delta p Gamma p = g with k 2 (0; 1] ....

Z. Chen, R. E. Ewing, and R. Lazarov, Domain decomposition algorithms for mixed methods for second order elliptic problems, Math. Comp. 65 (1996), 467-490.

....DIMENSIONS 189 difference with those in [GW87] MR94] and [CMW95] The Lagrange multipliers approach used in [GW87, CMW95] does produce a symmetric and positive definite matrix, but fails to address the other issue: the large number of unknowns. Recently, Chen, Ewing and Lazarov suggested in [CEL96] to reduce the number of unknowns by eliminating on a element by element basis the pressure variables. Then they applied a DD algorithm on the Lagrange multiplier variables only. When comparing the two basic ideas (i.e. Lagrange multipliers and div free subspace) one notes that they both ....

Chan Z., Ewing R. E., and Lazarov R. D. (1996) Domain decomposition algorithms for mixed methods for second-order elliptic problems. Math. Comp. 65: 467--490.

....be accomplished by using the Schur complement as in algorithms of Uzawa type, via the introduction of additional Lagrange multipliers and elimination of the vector variable, or via the use of divergence free bases and elimination of the scalar variable. Examples of such approaches can be found in [4, 6, 8, 9, 14, 15, 16, 17, 20, 22, 24, 32]. Finally, in this section we shall consider an application of the preconditioning of the k dependent operator associated to the bilinear form (6.1) Consider the system obtained by applying the mixed finite element method to the singular perturbation problem k 2 Delta p Gamma p = g ....

Z. Chen, R. E. Ewing, and R. Lazarov, Domain decomposition algorithms for mixed methods for second order elliptic problems, Math. Comp. 65 (1996), 467-490.

....grant DMS 9973328 and by the gift grant by Saudi Aramco Oil Co. The second author has been also supported by the Bulgarian NSF Grants MM 801. spaces (see, e.g. 1, 6] The work in this direction resulted also in construction of e#cient iterative methods for solving mixed FE systems (see, e.g. [7 9]) Galerkin method based on non conforming Crouzeix Raviart linear triangular finite elements has been also used in the construction of so called locking free approximations for parameter dependent problems. Furthermore, the sti#ness matrix has a regular sparsity structure such that in each row ....

Z. Chen, R.E. Ewing, and R.D. Lazarov, Domain decomposition algorithms for mixed methods for second order elliptic problems, Math. Comp., 65(214) (1996), 467-490.

.... 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 elements. We first consider the convergence of the W cycle and variable V cycle algorithms for these nonconforming elements. We prove ....

Zhangxin Chen, R. Ewing, and R. Lazarov, Domain decomposition algorithms for mixed methods for second order elliptic problems, Math. Comp. 65 (1996), to appear.

....Shauder fixed point theorem and a regularization technique. Moreover, the weak solution is approximated by using the mixed finite element method, which has the advantage of preserving the mass locally on each element. In addition, most of the techniques developed earlier by Ewing and colleagues [14, 9, 10, 15, 16, 17] for the mixed method have a strong potential in solving free or moving boundary value problems. The paper is organized as follows. In x2, we shall describe the model problem in simulation. In x3, we derive a weak formulation for the steadystate problem in mixed form. In x4, we prove an existence ....

....h which is located in the interior of the domain D. The mixed finite element method allows a natural use of locally refined grids in practical simulations. Therefore, most of the techniques which have A FREE BOUNDARY PROBLEM FOR FLUID FLOW IN POROUS MEDIA 9 been developed Ewing and colleagues [14, 9, 10, 15, 16, 17] are applicable to the free boundary value problem considered in this article. There are a couple of other open problems regarding the numerical scheme (5.1) First, the error between the true solution (p; u; and its approximation (p h ; u h ; h ) should be established. Second, the uniqueness ....

Z. Chen, R. E. Ewing, and R. D. Lazarov, Domain decomposition algorithms for mixed methods for second order elliptic problems, Math. Comp. 65(214) (1996), 467--490.

....to solve compared with the de nite systems. However, the popularity of the mixed method has increased considerably as a consequence of the progress made in recent years in developing ecient Mathematical and Numerical Techniques in Energy 9 algorithms for solving this inde nite system (see, e.g. [4, 9, 10, 19, 31, 32, 34, 56, 64, 65, 77]) The mixed method for approximating accurately the total velocity v is used in this paper. This method is further examined and exploited in the papers by Arbogast, Douglas Pereira Yeh, GaranzhaKonshin Lyons Papavassiliou Qin, Huang Spagnuolo, Ming Shi, Qin WangEwing Espedal, Wang Li, Russell, ....

Chen, Z., Ewing, R. E., and Lazarov, R., Domain decomposition algorithms for mixed methods for second order elliptic problems, Math. Comp. 65 (1996), 467-490.

....years, the study of multigrid methods for mixed finite element methods, which are popular in the simulation of fluid flow in porous media [21] has been quite active; see [2, 19, 31, 43, 46, 47] for example. However, due to the equivalence between nonconforming and mixed finite element methods [2, 3, 17, 19, 23], the analysis for the nonconforming finite methods directly applies to the mixed methods. Thus all the results derived here carry over to the mixed methods. Also, the present techniques can be used to analyze other nonconforming elements. The rest of the paper is organized as follows. In the next ....

Z. CHEN, R. E. EWING, AND R. LAZAROV,Domain decomposition algorithms for mixed methods for second order elliptic problems, Math. Comp., 65 (1996), pp. 467--490.

....years, the study of multigrid methods for mixed finite element methods, which are popular in the simulation of fluid flow in porous media [21] has been quite active; see [2, 19, 30, 42, 45, 46] for example. However, due to the equivalence between nonconforming and mixed finite element methods [2, 3, 17, 19, 23], the analysis for the nonconforming finite methods directly applies to the mixed methods. Thus all the results derived here carry over to the mixed methods. Also, the present techniques can be used to analyze other nonconforming elements. INTERGRID OPERATORS AND MULTIGRID METHODS 3 The rest of ....

Z. Chen, R. E. Ewing, and R. Lazarov, Domain decomposition algorithms for mixed methods for second order elliptic problems, Math. Comp., 65 (1996), 467--490.

....iterations. Not only is the analysis of multigrid algorithms for nonconforming finite element methods of interest for their own sake (see, e.g. 15] 17] 18] 21] and the bibliographies therein) but it has great application to mixed finite element methods. It has been shown [12] [14], 15] that the linear system arising from the mixed methods of the symmetric problem can be algebraically condensed to a symmetric, positive definite system for Lagrange multipliers. This linear system is identical to the system arising from the nonconforming finite element methods. Hence the ....

....for the nonsymmetric and indefinite problem, both types of multigrid methods mentioned above are tested for the first time. The later analysis is carried out for the two dimensional, triangular case; it works for the three dimensional case without substantial changes as noticed in [12] [14], 15] Also, rectangular finite elements can be similarly considered. ANALSIS OF MULTIGRID ALGORITHMS 3 2. The Symmetric Problem. In this section we develop multigrid algorithms for the symmetric problem. The nonconforming finite elements are considered in x2.1, and the mixed methods are ....

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Zhangxin Chen, R.E. Ewing, and R. Lazarov, Domain decomposition algorithms for mixed methods for second order elliptic problems, Math. Comp., 65 (1996), to appear.

....mentioned above, so in fact oe h 2 V h . In [2,27] it was shown that in the case of the lowest order Raviart Thomas elements the solution to (1.4) can be recovered from the Galerkin method which uses nonconforming linear elements augmented with bubble functions. In this paper, following [1,12,14], we show that the linear system generated by (1.4) can be algebraically condensed to a symmetric and positive definite system for the Lagrange multiplier h . It is then shown that this linear system can be obtained from the Galerkin method for nonconforming linear elements without any bubbles. ....

....his valuable comments and suggestions. The results of this paper are part of the Partnership in Computational Sciences (PICS) Project on Groundwater Contaminant Transport sponsored by the US Department of Energy under Grant #DE FG05 92ER25143. This research together with the results published in [1,6,7,14,17,23,35] have given the theoretical foundation to the solution methods which have been implemented in the PICS codes. A detailed description of procedures to construct preconditioners using nonoverlapping domain decomposition and domain decomposition on nonmatching grids is given in [23] ....

Z. Chen, R. Ewing, and R. Lazarov. Domain decomposition algorithms for mixed methods for second order elliptic problems. Math. Comp., 65:467--490, 1996.

.... 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 elements. We first consider the convergence of the W cycle and variable V cycle algorithms for these nonconforming elements. We prove ....

Zhangxin Chen, R. Ewing, and R. Lazarov, Domain decomposition algorithms for mixed methods for second order elliptic problems, Math. Comp. 65 (1996), to appear.

....rectangular elements is carried out in the fourth section. Finally, numerical experiments on the performance of the present approaches are given in the fifth section. The later analysis is carried out for the two dimensional case; it works for the three dimensional case without substantial changes [14, 16, 17]. NONCONFORMING MULTIGRID ALGORITHMS Problem (1.1) is recast in weak form as follows. We define the bilinear form a( Delta; Delta) as follows: a(v; w) arv; rw) v; w 2 H 1( Omega Gamma ; where ( Delta; Delta) denotes the L 2( Omega Gamma or (L 2( Omega Gamma8 2 inner product, as ....

....can establish the following stability property [6, 9, 14] a k (I k v; I k v) Ca k Gamma1 (v; v) 8 v 2 V k Gamma1 ; 2:5) with C independent on k, the constant C is in general bigger than two, as observed in [18] Example 2. The second example was originally described in [33] and then used in [16] for analyzing domain decomposition methods for mixed finite element methods. If v 2 V k Gamma1 and E 2 E k Gamma1 with the vertices q i and the midpoints q i of its edges, i = 1; 2; 3, then I k k Gamma1 v(q i ) v(q i ) i = 1; 2; 3; I k k Gamma1 v(q i ) 1 N1 P j v(q 0 j ) if q i = 2 ....

[Article contains additional citation context not shown here]

Z. Chen, R. Ewing, and R. Lazarov, Domain decomposition algorithms for mixed methods for second order elliptic problems, ISC-94-06-MATH Technical Report, Texas A&M University.

....is slightly more difficult. Not only is the analysis of multigrid algorithms for nonconforming finite element methods of interest for their own sake (see, e.g. 10] 14] 15] 16] and the bibliographies therein) but it has application to mixed finite element methods. It has been shown [8] [9], 10] that the linear system arising form the mixed methods can be algebraically condensed to a symmetric, positive definite system for Lagrange multipliers. This linear system is identical to the system arising from the nonconforming finite element methods. Hence the analysis of multigrid ....

....presented here and comparing the nonconforming multigrid methods with standard conforming finite element and finite difference multigrid methods. The later analysis is carried out for the two dimensional case; it works for the three dimensional case without substantial changes as noticed in [8] [9], 10] We end this section with three remarks. First, a lower order term is included in (1.1) so our results also apply to parabolic problems. Second, the general theory for the analysis of multigrid methods developed in [5] has application to conforming ANALSIS OF MULTIGRID ALGORITHMS 3 ....

[Article contains additional citation context not shown here]

Z. Chen, R.E. Ewing, and R. Lazarov, Domain decomposition algorithms for mixed methods for second order elliptic problems, ISC-94-06-MATH Technical Report, Texas A&M University, submitted for publication.

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Z. Chen, R. E. Ewing, and R. Lazarov. Domain decomposition algorithms for mixed methods for second order elliptic problems. Math. Comp., 65:467-490, 1996.

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Z. Chen, R. E. Ewing, and R. Lazarov. Domain decomposition algorithms for mixed methods for second order elliptic problems. Math. Comp., 65:467-490, 1996.

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Z. Chen, R. E. Ewing and R. Lazarov, Domain Decomposition algorithms for mixed methods for second-order elliptic problems, Math. Comp., 65(214), pp.467-490, April 1996.

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