Glowinski, R. and Wheeler, M.F.: Domain decomposition and mixed finite element methods for elliptic problems, presented in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. Periaux, eds., SIAM, Philadelphia, 1988, pp. 144-172.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....choices for clumping elements also lead to simple codes. Although designed for distributed memory machines, the above properties make the new method competitive even when used in serial machines. Domain decomposition techniques for mixed methods distinct from the one used here can be found in [13, 30, 29, 34, 35]. Other Eulerian Lagrangian approaches leading to conservative schemes different from ours can be found in [2, 4, 8, 9, 37] A brief history of forerunners of the technique given in [23] and here can be found in [23] The model problem we have taken to test our numerical method corresponds ....

R. Glowinski and M. F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in "Domain Decomposition Methods for Partial Differential Equations", 144-172 (R. Glowinski, G. Golub, G. Meurant, J. Periaux, eds.), SIAM, Philadelphia (1988).

.... and superconvergence estimates can be found, for example, in [19, 8, 16, 23, 10, 11] Since the standard MM yields a linear system that represents a saddle point problem, much current research on the MM involves how to efficiently solve the system of equations that arises, see for example, [13, 5, 6, 12, 21, 17, 1]. Perhaps the earliest successful technique was the hybrid form of the mixed method (HM) 3] This turns the saddle point problem into a semi definite problem, but at the expense of greatly increasing the number of unknowns. This research was supported in part by the Department of Energy, the ....

....added at edges where g is not smooth. In the HM, Lagrange multipliers were introduced on the boundary of every element in Th. In some cases, Lagrange multipliers are only needed on the boundaries of a few elements. For example, when applying the domain decomposition techniques described in [13], Lagrange multipliers are introduced only on the element edges where sub domains intersect. In this method, just as in (33) one can reduce the global system of equations to an equation for the Lagrange multipliers. Applying the resulting matrix operator involves solving sub domain problems for ....

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R. Glowinski and M. F. Wheeler, Domain decomposition and mixed finite element methods for elliptic pro blems, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. Meurant, J. Periaux, eds., SIAM, Philadelphia, pp. 144-172, 1988.

....appropriate parallel computing platforms. Actually, the scope of methodology ranges from the theory of partial differential and integral equations, numerical mathematics, and parallel computation to the mathematical modeling and numerical simulation of complex technological processes (cf. e.g. [1 4, 11 14, 16, 17, 20 22, 24, 28, 29 32, 38, 40, 42 45]) A new powerful approach in this class of methods has recently been provided, which has become known as domain decomposition on nonmatching grids (cf. 1 4, 11 13, 16, 21, 22, 28 30, 38, 42 45] This approach stems from the macrohybrid formulations of differential problems with Lagrange ....

R. Glowinski and M. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems. In: Domain Decomposition Methods for Partial Differential Equations (Eds. R. Glowinski et al.). SIAM, Philadelphia, 1988, pp. 144 -- 172.

....and avoid the inversion of A. There is also a variety of application specific techniques that depend strongly on the particular approximation spaces, geometry of the domain etc. In the case of the mixed approximation of second order problems, those include domain decomposition techniques [17], a reduction technique involving the use of additional Lagrange multipliers [9] as well as an indefinite preconditioner [13] The inexact Uzawa algorithms are of interest because they are simple and have minimal computer memory requirements. This could be important in large scale scientific ....

R. Glowinski and M. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in First Inter. Symp. on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. Golub, G. Meurant, and J. Periaux, eds., Phil. PA, 1988, SIAM, pp. 144--172.

.... (BDD) was introduced by Mandel [19] by adding a coarse problem to an earlier method of De Roeck and Le Tallec [11] known as the Neumann Neumann method, based in turn on earlier work for the case of two subdomains [2] and on a closely related method of Glowinski and Wheeler for mixed problems [17]. The development of BDD was motivated by very good performance of the Neumann Neumann preconditioner for real world problems with strongly discontinuous coefficients for a small number of subdomains [11] An algorithm similar to BDD but different in important aspects and convergence results also ....

R. Glowinski and M. F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. P'eriaux, eds., Philadelphia, PA, 1988, SIAM.

....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] ....

R. Glowinski and M. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski et. al., eds., SIAM Publications, Philadelphia, 1988, pp. 144-172.

....to characterize the divergence free velocity subspaces. This is also the essential Ninth International Conference on Domain Decomposition Methods Editor Petter E. Bjrstad, Magne S. Espedal and David E. Keyes c fl1998 DDM.org DD FOR A MIXED FEM IN THREE 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 ....

....essential Ninth International Conference on Domain Decomposition Methods Editor Petter E. Bjrstad, Magne S. Espedal and David E. Keyes c fl1998 DDM.org DD FOR A MIXED FEM IN THREE 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 ....

[Article contains additional citation context not shown here]

Glowinski R. and Wheeler M. F. (1987) Domain decomposition and mixed finite element methods for elliptic problems. In Glowinski R., Golub G. H., Meurant G. A., and P'eriaux J. (eds) Proc. 1st Int. Symp. on Domain Decomposition Methods. SIAM, Philadelphia.

....on Gamma 1 ; 2 = Gammaq 1 u 1 n 1 on Gamma 2 ; 3 = Gamma u 1 n 1 on Gamma 3 ; 2.15) where n 1 and n 2 are the outer normal vectors to Omega 1 and Omega 2 , respectively. Recall that u 1 j u in Omega 1 and u 2 j u in Omega 2 . In a compact form (2. 14) can be presented [GW88, BF91] by: find (u; 2 V Theta such that a(u; v) b( v) l(v) b( u) 0; 8(v; 2 V Theta : 2.16) Here V = m Q k=1 V k ; V k = H 1( Omega k ) k = 1; m; sg Q s=1 s ; s = H Gamma1=2 ( Gamma s ) s = 1; s g ; 2.17) a(u; v) m P k=1 a k (u; ....

Glowinski R. and Wheeler M. (1988) Domain decomposition and mixed finite element methods for elliptic problems. In Glowinski R., Golub G., G.Meurant, and Periaux J. (eds) Proc. First DD Int. Conf., pages 144--172. SIAM, Philadelphia.

....methods have received a lot attention during the last few years, due to the restrictions of overlapping domain decomposition methods. Several families of non overlapping decomposition methods for the solutions of elliptic problems have been proposed, analyzed, and successfully implemented [BW86, BPWX91, DPLRW93, Dry89, GW88, Lio90, MQ89, LTDRV91, Tan92]. In a non overlapping domain decomposition method, the original problem is first decomposed into smaller problems defined on non overlapping subdomains. Parallel or sequential iterative procedures are then constructed for decoupling the whole domain problem into subdomain problems. During the ....

Glowinski R. and Wheeler M. F. (1988) Domain decomposition and mixed finite element methods for elliptic problems. In Proc. First International Symposium on Domain Decomposition Method for Partial Differential Equations. SIAM, Philadelphia.

....problem for the Lagrange multipliers is solved by a preconditioned conjugate gradient (PCG) algorithm. The FETI method was developed in [Far91, FR91, FR92] and discussed in detail in the monograph [FR94] Unlike other related domain decomposition methods using Lagrange multipliers as unknowns [GW88, Rou90], the FETI method uses the null spaces of the subdomain stiffness matrices (rigid body modes) to construct a small coarse problem that is solved in each PCG iteration. It was recognized in [FMR94] and proved mathematically in [MT96] that solving this coarse problem accomplishes a global exchange ....

Glowinski R. and Wheeler M. F. (1988) Domain decomposition and mixed finite element methods for elliptic problems. In Glowinski R., Golub G. H., Meurant G. A., and P'eriaux J. (eds) First International Symposium on Domain Decomposition Methods for Partial Differential Equations, pages 144--172. SIAM, Philadelphia.

....is also possible, at least formally, to use the same SRC transmision conditions to domain decompose in time. DDM FOR CONTROL PROBLEMS 271 6 Remarks Numerical Implementation and Speed of Convergence A good way of discretizing these problem is to use mixed hybrid finite elements (see [CJ86] or [RG90a, RG88] on the use of mixed finite elements in domain decomposition methods) This approach is well suited to our problem for it uses in particular, as degrees of freedom, the fluxes of the normal derivatives and the average values of the trace of the direct and adjoint states on the interfaces which are ....

R. Glowinski M. W. (1988) Domain decomposition and mixed finite element methods for elliptic problems. In Glowinski R., Golub G. H., Meurant G. A., and P'eriaux J. (eds) First international symposium on domain decomposition methods for partial differential equations. SIAM.

....MIXED SYSTEMS 449 introducing Langrange multipliers on all the element edges. For a general discussion of the hybrid mixed finite element method we refer to Brezzi and Fortin [15] The development of domain decomposition preconditioners for the hybrid method is discussed by Glowinski and Wheeler [25], Cowsar [16] and Cowsar, Mandel and Wheeler [17] In Rusten and Winther [32] an alternative to the standard hybrid version of the mixed method is discussed, where the continuity requirements are only relaxed on the edges of a coarse grid. This relaxation has the e#ect that the operator L becomes ....

R. Glowinski and M. F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, Proceedings, First International Conference on Domain Decomposition Methods (Philadelphia) (R. Glowinski et al., eds.), SIAM, 1988, pp. 144--172. MR 90a:65237

....Second Order Transmission Conditions for the Helmholtz Equation Jim Douglas, Jr. and Douglas B. Meade 1 Introduction Several domain decomposition methods for the solution of elliptic problems have been proposed, analyzed, and successfully implemented during the past decade [BW86, BPS86, GW88, HTJ88, Lio88, Lio90]. In recent years these ideas have been extended to non elliptic equations such as the Helmholtz equation [Des91, Des93, Des95, Ben95] and the harmonic Maxwell system [DJR92] It is well known [Lio88] that iterative methods using the Dirichlet or Neumann transmission conditions may not converge to ....

Glowinsky R. and Wheeler M. F. (1988) Domain decomposition and mixed finite element methods for elliptic problems. In Glowinski R., Golub G., Meurant G., and P'eriaux J. (eds) Proc. First Int. Syposium on Domain Decomposition Methods for Partial Differential Equations, pages 144--171. SIAM.

....by Farhat, Mandel, and Roux a few years later [14] Our own work is based on the pioneering work by Mandel and Tezaur [24] who fully analyzed a variant of that algorithm. For a detailed introduction, see [15] or [33] For early work on the Neumann Neumann algorithms and their predecessors, see [5, 18, 2, 9, 3, 4, 22]. For a fine introduction, see [21] The purpose of this paper is to extend, simplify, and unify the theory for the FETI and Neumann Neumann algorithms. We introduce a new one parameter family of FETI preconditioners and prove a bound on the rate of convergence which is independent of possible ....

Roland Glowinski and Mary F. Wheeler. Domain decomposition and mixed finite element methods for elliptic problems. In Roland Glowinski, Gene H. Golub, Gerard A. Meurant, and Jacques Periaux, editors, First International Symposium on Domain Decomposition Methods for Partial Di#erential Equations, Philadelphia, PA, 1988. SIAM.

....partial differential equations. Recently, the author studied finite difference and finite element DD methods for solving complex valued scalar waves without introducing Lagrange multipliers [32, 34, 35] DD techniques for flows in porous media distinct from the one suggested here can be found in [7, 27, 30, 31]. 4.1. A nonoverlapping DD method Let f Omega j #j=1# Delta Delta Delta#Mg be a nonoverlapping partition of Omega; Omega; M [ j=1 Omega j # Omega j Omega k = ## j 6= k: Assume also that each Omega j is a (logically) cubic region. The method to be presented is equally applicable to ....

R. Glowinski and M. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. Golub, G. Meurant, and J. Periaux, eds., Philadelphia, 1988, SIAM, pp. 144--172.

....the Lagrange multipliers, and which is best solved by a preconditioned conjugate gradient (PCG) algorithm. The FETI method was developed in [8, 13, 14] and discussed in detail in the monograph [15] In contrast with other related domain decomposition methods using Lagrange multipliers as unknowns [17, 27], the FETI method distinguishes itself with the treatment of the null spaces of the subdomain stiffness matrices (rigid body modes) associated with the so called floating subdomains, i.e. subdomains without a sufficient number of essential boundary conditions to prevent singularities of the local ....

R. Glowinski and M. F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in First International Symposiumon Domain DecompositionMethods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. P'eriaux, eds., Philadelphia, 1988, SIAM, pp. 144--172.

.... and further studied in [12, 28, 29] Our abstract framework and analysis are related to the representation of another Neumann Neumann method with a coarse space as an abstract Schwarz method in [16] For earlier work on the Neumann Neumann and similar preconditioners without a coarse space, see [1, 13, 14, 21, 32]. A domain decomposition method in a sense dual to Neumann Neumann is obtained by enforcing intersubdomain continuity by Lagrange multipliers [19, 35, 17] That method, known as FETI, gives rise to a natural coarse problem [18, 30] A version of FETI with convergence properties for plates similar ....

R. Glowinski and M. F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. P'eriaux, eds., Philadelphia, PA, 1988, SIAM.

....partial differential equations. Recently, the author studied finite difference and finite element DD methods for solving complex valued scalar waves without introducing Lagrange multipliers [32, 34, 35] DD techniques for flows in porous media distinct from the one suggested here can be found in [7, 27, 30, 31]. 4.1. A nonoverlapping DD method Let f Omega j ; j = 1; Delta Delta Delta ; Mg be a nonoverlapping partition of Omega Gamma Omega = M [ j=1 Omega j ; Omega j Omega k = j 6= k: Assume also that each Omega j is a (logically) cubic region. The method to be presented is equally ....

R. Glowinski and M. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. Golub, G. Meurant, and J. Periaux, eds., Philadelphia, 1988, SIAM, pp. 144--172.

....by (6.25) satisfies #(T b S) # log 2 h 0 h . Additional bibliography remarks. The methods studied in this section, often known as Neumann Neumann type algorithms, can be traced back to the work by Dinh, 896 JINCHAO XU AND JUN ZOU Glowinski, and Periaux [33] and Glowinski and Wheeler [46]. Thereafter there are a few extensions in the theory and algorithms. We refer to Bourgat et al. 7] Roeck and Le Tallec [67] Le Tallec, Roeck, and Vidrascu [76] Mandel and Brezina [58, 59] and Dryja and Widlund [40] For extension of the approach for mixed finite element framework by ....

....there are a few extensions in the theory and algorithms. We refer to Bourgat et al. 7] Roeck and Le Tallec [67] Le Tallec, Roeck, and Vidrascu [76] Mandel and Brezina [58, 59] and Dryja and Widlund [40] For extension of the approach for mixed finite element framework by Glowinski and Wheeler [46] to the many subdomain case, see Cowsar and Wheeler [32] The weighted coarse subspace originated from Bramble, Pasciak, and Schatz [11, 12] for solving their coarse problem, and was used by Dryja and Widlund [40] and Mandel [58] for the Neumann Neumannn type methods. In [40] the use of standard ....

R. Glowinski and M.F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in First International Symposium on Domain Decomposition Methods for Partial Di#erential Equations, R. Glowinski, G.H. Golub, G.A. Meurant, and J. Periaux, eds., SIAM, Philadelphia, PA, 1988, pp. 144--172.

....domain # , is decomposed into a series of subdomains (faultblocks, blocks) # k , k 1, n b .Let# kl ## l be the interface between # k and # l . A physical model is associated with each block. This domain decomposition formulation for mixed methods stems from the classical paper [24] and was extended to the non matching grids with mortar spaces in [3] 3.1. Interface and boundary conditions On each interface # kl the following physically meaningful interface continuity conditions are imposed: # on # kl , 10) UM #] kl 0on# kl , 11) where # k denotes the ....

R. Glowinski and M.F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in: Domain Decomposition Methods for Partial Differential Equations (SIAM, Philadelphia, PA, 1988) pp. 144--172.

....local refinements to capture the spatial behavior of the solution. In that case, non overlapping domain decomposition techniques with Mortar elements at the interfaces of the decomposition have proven to be e#cient since they enable to define the grids independently in the subdomains regions (see [GW88], Yot96] ACWY96] On the other hand, the transient behavior of the solution may also warrant the use of di#erent time steps in the di#erent subdomains. The idea of the domain decomposition method introduced in this paper is to combine Mortar Mixed Finite Element methods for the space ....

....## 1 2 ,# : sup v#V P N i=1 R # i (vn i )d# #v#V , and we shall denote by H 1 2 (#) the subspace of L 2 (#) of functions such that ## 1 2 ,# #. We consider, on the domain decomposition(# i ) i=1, N , a Mortar Mixed Finite Element (MMFE) discretization of (1) introduced in [GW88] for matching grids, and extended in [Yot96] ACWY96] to the case of non matching grids at the interfaces between the subdomains# i . In that case, a so called Mortar space # h # L 2 (#) is introduced on the skeleton #. Then, equation (1) is discretized on each subdomain by a Mixed Finite ....

Roland Glowinski and Mary F. Wheeler. Domain decomposition and mixed finite element methods for elliptic problems. In Roland Glowinski, Gene H. Golub, Gerard A. Meurant, and Jacques Periaux, editors, First International Symposium on Domain Decomposition Methods for Partial Di#erential Equations, Philadelphia, PA, 1988. SIAM.

....et al. 1992; Arbogast and Wheeler, 1995, Dawson, 1991) The coupled flow and transport equations are solved using operator splitting. First, the mass conservation mixed finite element approximation for flow is solved implicitly for velocity using the Glowinski Wheeler domain decomposition method (Glowinski and Wheeler, 1988) in parallel. This step is independent of the transport calculation and may have an independent time step size. The computed velocity field is passed to the transport calculation, which solves for concentrations of mobile and immobile species. This step is done independently using a time splitting ....

Glowinski, R. and Wheeler, M. F., 1988, Domain decomposition and mixed finite element methods for elliptic problems, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, pp.144-172.

No context found.

Glowinski, R. and Wheeler, M.F.: Domain decomposition and mixed finite element methods for elliptic problems, presented in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. Periaux, eds., SIAM, Philadelphia, 1988, pp. 144-172.

No context found.

R. Glowinski and M. F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in: R. Glowinski, G. Golub, G. Meurant, and J. P'eriaux, Eds., Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1988, pp. 144-- 172.

No context found.

R. Glowinski and M. F. Wheeler. Domain decomposition and mixed finite element methods for elliptic problems. In R. Glowinski, G. Golub, G. Meurant, and J. P'eriaux, editors, Proc. First SIAM Conference on Domain Decomposition Methods for Partial Differential Equations, pages 144--172. SIAM, 1988.

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