J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang. A parallel iterative procedure applicable to the approximate solution of second order partial dierential equations by mixed nite element methods. Numer. Math., 65 (1993) 95-108.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the function and its normal derivative on the smoothing Robin interface conditions. This method was rst proposed and analyzed in [10] where, through energy estimates, the convergence of the method at di erential level was established for arbitrary decompositions and elliptic operators. Later in [5,8], this method was further analyzed at discrete level in a nite element framework. Several variations of this method have been also appeared. In [7] an ADI based modi cation is considered and analyzed at discrete level for model problems and decompositions. A second variation of the ROB method ....

J. Douglas Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang. A parallel iterative procedure applicable to the approximate solution of the second order partial dierential equations by mixed nite element methods. Numer. Math., 65:95{ 108, 1993. 26

....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 ....

....be transmitted between subdomains in order to guarantee convergence. This information transmission step is the key part of a domain decomposition method; it distinguishes one domain decomposition method from another. Several methods for passing information have been proposed in the literature [BF96, DPLRW93, GLT90, Lio90, MQ89]. The common approach was to develop a transmission condition for the differential problem and adapt the same condition to the corresponding discrete problem. The purpose of this paper is to present a parallelizable, iterative, non overlapping domain decomposition method for solving second oreder ....

[Article contains additional citation context not shown here]

Douglas Jr. J., Paes Leme P. J. S., Roberts J. E., and Wang J. (1993) A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods. Numer. Math. 65: 95--108.

....improved variant of Lions s method is proposed by Q. Deng and its convergence is analyzed in the Sobolev H 1 norm [Den97] Exploiting the structure of mixed finite element, Douglas et al. obtained a more precise convergence rate by a spectral radius estimation of the iterative solution operator [DPRW93]. More e#cient iterative schemes, such as Seidel type and under relaxation type domain decomposition iterative methods for elliptic, Helmholtz and electromagnetic problems have been considered in [CGJ98, CDJP97, DM97, Fen97, Gha97] and Seidel type approaches based on nonconforming finite elements ....

J. Douglas, Jr., P. L Paes Leme, J. E. Roberts, and J. Wang. A parallel iterative procedure applicable to the approxiamte solution of second order partial di#erential equations by mixed finite element methods. Numer. Math., 65:95--108, 1993. 170 KWON, SHEEN

....of the function and or its normal derivative on the smoothing Robin interface conditions. This method was first analyzed in [8] where, through energy estimates, the convergence of the method at differential level has been established for arbitrary decompositions and elliptic operators. Later in [2,6] this method was been further analyzed at discrete level in a finite element framework. Several variations of this method have been also appeared. In [5] an ADI based modification is considered and analyzed at discrete level for model problems and decompositions. A second variation of ROB method ....

J. Douglas Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang. A parallel iterative procedure applicable to the approximate solution of the second order partial differential equations by mixed finite element methods. Numer. Math., 65:95-- 108, 1993.

....; k = 1; 2; Delta Delta Delta ; K; m 6= k; 12) 1 u n 1 1 j 1 u n 1 1 = g a ; x = a; 13) 2 u n 1 K j 2 u n 1 K = g b ; x = b; 14) where km is a parameter which can be chosen to speed up the convergence, and L is the given elliptic operator. Despr es [1] Douglas et al. [2], and Kim [4] have considered discretizations of this procedure by the hybridized mixed finite element method and the finite difference method and applied the procedure to wave equations. Mu and Rice [7] considered collaborating PDE solvers in the context of domain decompositions. 2.2 The finite ....

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts and J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods, Numer. Math., 65(1993), pp. 95-108.

....This method allows an arbitrary decomposition of the domain and each subdomain problem plays the same role in computation. However, the convergence of this method is very slow unless some parameter is carefully chosen. The idea of this method has been used by Despr es [6] and Douglas et al. [9] to construct mixed finite element domain decomposition methods. In Rice, Vavalis and Yang [21] and Yang [31] Dirichlet data are passed at odd iterations and Neumann data at even iterations. This method requires that the information at the previous iteration level be passed to the subdomain ....

....be 1 2 . Although our convergence analysis is made at the differential level, finite dimensional approximations are easy to consider. In particular, we may apply the finite element method (with or without Lagrange multipliers) and mixed finite element methods. The analysis should be similar to [9, 11, 31] It appears that our method may not apply directly to domain decompositions with corner points or cross points. Thus the number of subdomains should be small, since large number of subdomains will lead to very long and narrow subdomains. Although some problems do not require a large number of ....

Douglas, J. Jr., Paes Leme, P. J., Roberts, J. E., & Wang, J. 1993 A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods. Numer. Math. 65, 95-108.

....k k kmu n Gamma1 k on Gamma mk ; k = 1; 2; Delta Delta Delta ; M; m 6= k; 6) u n m = g; x 2 Omega ; 7) where km is a parameter which can be chosen to speed up the convergence, and m is the outward normal to the boundary of subdomain Omega m . Despr es [1] Douglas et al. [2], and Kim [3] have considered discretizations of this procedure by finite element methods and applied the procedure to wave equations. Mu and Rice [5] considered collaborating PDE solvers in the context of domain decompositions. When imposing the Robin boundary condition (6) weakly and Dirichlet ....

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts and J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods, Numer. Math., 65 (1993), pp. 95-108.

.... Maxwell equations [20] The method has inspired more sophisticated numerical algorithm such as conjugate gradient like iterative algorithm [14] 18] and a new weak formulation of the Helmholtz equation [11] The method is easily applied to dispersive media [25] and coercive elliptic equations [17] [26] [3] Let us finally mention the existence of a related work in [38] The extension to optimal control problem described in this paper has been used to solve 3 D acoustic problems [5] Its application to electromagnetic active control is under investigation. The application of these techniques to ....

J. E. Roberts J. Wang J. Douglas Jr., P.J. Paes Leme. A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods. Numer. Math., 65:95--108, 1993.

....ku h 0 k 0;1 ; oe 0: For the example considered in this section, D(10) 6:3 and D(20) 11:5 . A block Jacobi type nonoverlapping DD method of a Robin interface condition is first considered to solve the problem (3) The scope of the DD method and its various applications can be found in [5, 7, 13, 15, 16]. For the case of 8 Theta 8 subdomains, the 4 0 1 2 3 4 5 0 1 2 3 4 5 0.2 0.15 0.1 0.05 0 0.05 x(Km) y(Km) Figure 2: The difference (u h 0 Gamma u h 20 ) DD method takes 213 iterations for oe = 0, 59 iterations for oe = 10, and 38 iterations for oe = 20. We then ....

J. Douglas, Jr., P.J. Paes Leme, J.E. Roberts, and J. Wang. A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods. Numer. Math., 65:95--108, 1993.

....a RIBC was first analyzed by Lions [37] where the convergence of the algorithm was proved in differential, rather than discrete, level. Hybrid mixed finite element DD methods introducing Lagrange multipliers were employed by Despr es [8] for solving the Helmholtz problem, and Douglas et al. [18] analyzed the convergence of the method for symmetric second order 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 ....

....= 0; x 2 Gamma j ; d) Gammafi u j Delta j p j = fiu k Delta k p k ; x 2 Gamma jk ; 4.6) where j is the unit outer normal to Omega j , and fi is a positive function defined on the interfaces. The RIBC (4. 6d) is equivalent to (and more convenient than) the consistency conditions [8, 18, 34, 37] p j = p k ; u j Delta j u k Delta k = 0; on Gamma jk : 4.7) Now, move toward a new weak formulation by testing (4.6a) against a vector v 2 V j = Vj Omega j : D Gamma1 u j ; v) Omega j Gamma (p j ; r Delta v) Omega j hp j ; v Delta j i Omega j = 0; v 2 V j : 4.8) ....

[Article contains additional citation context not shown here]

J. Douglas, Jr., P. Paes Leme, J. Roberts, and J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods, Numer. Math., 65 (1993), pp. 95--108.

....parameter is given. As an aside in his thesis on the application of this technique to the more difficult Helmholtz problem, Despr es [1] extended Lions s proof to a more general partition but again without an estimate of an explicit rate of convergence. Douglas, Paes Leme, Roberts, and Wang [8] did obtain convergence rates for fixed fi for a mixed finite element approximation of the elliptic boundary problem under a number of different hypotheses on the coefficients and the partition. Motivated by the ADI method, I co authored a paper [3] with Jim Douglas, Jr. that studies a version ....

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang. A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods. Numer. Math., 65 (1993) 95--108.

....(1991) In both Lions (1990) and Despr es (1991) the domain decomposition methods were analyzed only for the differential problems; it is not trivial to apply this idea to the corresponding discrete problems. For the coercive elliptic problems, two successful approaches were reported recently in Douglas, Paes Leme, Roberts Wang (1993) and Le Tallec Sassi (1995) respectively. Douglas, Paes Leme, Roberts Wang (1993) proposed a discrete version of the Robin transmission condition for mixed finite element methods by making strong use of hybridization of mixed finite element methods. Convergence and the rate of convergence for ....

....analyzed only for the differential problems; it is not trivial to apply this idea to the corresponding discrete problems. For the coercive elliptic problems, two successful approaches were reported recently in Douglas, Paes Leme, Roberts Wang (1993) and Le Tallec Sassi (1995) respectively. Douglas, Paes Leme, Roberts Wang (1993) proposed a discrete version of the Robin transmission condition for mixed finite element methods by making strong use of hybridization of mixed finite element methods. Convergence and the rate of convergence for the domain decomposition method were established. Le Tallec Sassi (1995) presented ....

[Article contains additional citation context not shown here]

Douglas, Jr., J., Paes Leme, P. J. S., Roberts, J. E., & Wang, J. (1993), A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods, Numer. Math. 65, 95--108.

.... the Galerkin approximations in both [16] and [17] The domain decomposition methods developed in this paper are based on the idea of using the convex combinations of the interface conditions in place of the original interface conditions to pass the information between subdomains, see [11] 2] [6], 9] and references therein for the expositions and discussions on this approach for problems posed in homogeneous media. So the domain decomposition methods of this paper may be regarded as the generalizations of the methods proposed in those papers to the time dependent heterogeneous problems. ....

....interface problems arising from many scientific applications. We refer to [14] and reference therein for more discussions in this direction. For simplicity we shall only describe and analyze our domain decomposition algorithms at the differential level in this section. Following the ideas of [2] [6] and [9] it is not very hard but rather technical and tedious to construct and analyze the discrete analogues of the differential domain decomposition algorithms to be introduced in the following. Those analyses along with the computation test results will be reported elsewhere in a forthcoming ....

J. Douglas, Jr, P. L. S. Paes Leme, J. E. Roberts and J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods, Numer. Math. 65 (1993), 95--108.

....similar to their overlapping counterpart, but require that transmission conditions on the interface be designed carefully to ensure the convergence. Examples are Funaro, Quarteroni, and Zanolli [19] Marini and Quarteroni [32, 33] Lions [31] Despr es [14] Douglas, Paes Leme, Roberts, and Wang [15], Kim [23] Quarteroni [34] Le Tallec and Tidriri [25] Benamou and Despres [3] Engquist and Zhao [18] and Yang [40, 41, 42] In these methods, Robin or alternating DirichletNeumann interface conditions are applied. These methods are especially desirable for the so called interface or ....

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts and J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods, Numer. Math. 65(1993) 95-108. Domain Decomposition Methods for Elliptic Problems 19

No context found.

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang. A parallel iterative procedure applicable to the approximate solution of second order partial dierential equations by mixed nite element methods. Numer. Math., 65 (1993) 95-108.

.... competitiveness of the usual MMOC (see [26, 42, 28, 23] ffl Hybridized mixed finite elements for accurate velocity field computation (see [16, 17, 28] ffl A new domain decomposition iteration based on parallel iterative procedures introduced by [14, 15, 38] and the authors and collaborators [20, 21, 25] for solving the algebraic problems arising at each time step of a simulation. The resulting numerical scheme has several distinctive properties: Department of Mathematics, Purdue University, West Lafayette, IN 47907 1395, U.S.A. Department of Mathematics, University of Campinas, 13081 970 ....

....In the saturation iteration in the diffusive stage calculation, let us make the corresponding choice for sfi : sfi = Gamma c s h where c s is again a (dimensionless) constant. Then, c s so that (55) reduces to OE c s k fi = k k fi ) 1 2 Sn 1 : 58) In [21], convergence of domain decomposition iterations based on updating the Lagrange multipliers gfi and sfi in the mixed method formulations (31) 32) and (41) 42) for the global pressure p and the saturation s, respectively, was proved when the parameters gfi and sfi were assumed to be ....

[Article contains additional citation context not shown here]

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods, Numer. Math., 65 (1993) 95--108.

....will be used to represent the saturations of (many) other matrix blocks that lie inside the same cell. Gravity segregation within each computational cell is considered. The iterative procedures of [19] and this paper were motivated by [10, 11, 33] and previous work of the authors and collaborators [23, 24, 25, 26]. This paper is organized as follows. In x2, a detailed description of the medium block model is given along with some remarks which indicate our strategy for discretizing the space and time variables in the governing ow equations in both the fractures and the matrix blocks. Next, in x3 a ....

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial dierential equations by mixed nite element methods, Numer. Math., 65 (1993) 95-108.

....parameter is given. As an aside in his thesis on the application of this technique to the more dicult Helmholtz problem, Despr es [1] extended Lions s proof to a more general partition but again without an estimate of an explicit rate of convergence. Douglas, Paes Leme, Roberts, and Wang [2] did obtain convergence rates for xed for a mixed nite element approximation of (1) under a number of di erent hypotheses on the coecients and the partition. The paper is organized as follows. In x2, we consider the di erential model problem in the plane and indicate how its analysis carries ....

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang. A parallel iterative procedure applicable to the approximate solution of second order partial dierential equations by mixed nite element methods. Numer. Math., 65 (1993) 95-108.

....from a subdomain to its neighbors will be introduced for these methods. Quite analogous iterative procedures for conforming methods for second order elliptic problems were introduced rst by P. L. Lions [10, 11] and then applied to the more dicult Helmholtz problem by Despr es [7] later [8], a more precise convergence argument was established for the second order elliptic problem as approximated by mixed nite element methods. We shall analyze the convergence of the iteration for the nonconforming Galerkin method based on rectangular elements, using arguments related to those of [8] ....

....[8] a more precise convergence argument was established for the second order elliptic problem as approximated by mixed nite element methods. We shall analyze the convergence of the iteration for the nonconforming Galerkin method based on rectangular elements, using arguments related to those of [8]. Both two dimensional and three dimensional problems are discussed. The analysis would apply equally to nonconforming methods based on P 1 elements over simplices. The two dimensional case of the nite element method is hybridized in x11, and the domain decomposition procedure is de ned in x12. A ....

[Article contains additional citation context not shown here]

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang. A parallel iterative procedure applicable to the approximate solution of second order partial dierential equations by mixed nite element methods. Numer. Math., 65 (1993) 95-108.

....methods for coercive second order elliptic problems were introduced by Douglas et al. 12] and were based on ideas for conforming methods for second order elliptic problems introduced rst by P. L. Lions [16, 17] and then applied to the more dicult Helmholtz problem by Despr es [8] later [9], a more precise convergence argument was given for the coercive second order elliptic problem as approximated by mixed nite element methods. The organization of the paper is as follows. In x2 our model problem is stated. In x3 a global nonconforming Galerkin method is de ned and optimal order ....

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang. A parallel iterative procedure applicable to the approximate solution of second order partial dierential equations by mixed nite element methods. Numer. Math., 65 (1993), pp. 95-108.

....pj ; i 1 j ) h j ( i 1 pj ; i 1 j ) The mass fraction i 1 is calculated by the procedure described in x5.3 below. The nonlinear system (39) 40) is solved by a Newton method (combined with domain decomposition, leading to a naturally parallelizable iterative procedure; see [1, 10, 13]) 13 5.3 Di usion The di usive stage of the time step for the mass fraction is based on the approximation n 1 ( n 1 n 1 n 1 t d ; where n 1 is to be calculated at the transport stage described in x5.4. The equations to be solved for n 1 and d n 1 on each ....

....( j n;j n;j t d b n 1;j S n 1;j )h jx h jy ; b n 1;j = j n 1;j t d S n 1;j )h jx h jy ; j = 1; N: 14 5.3. 2 The implicit solution for the mass fractions The di usive stage system (41) 42) is solved for n 1 and d n 1 by the domain decomposition technique cited above [1, 10, 13]. The system is solved twice for two di erent estimates, n 1 ; 1; 2, resulting from two di erent approximations of the transported mass fraction, n 1 . Typically, one will underestimate and the other overestimate the solvent mass at time t n 1 . This will allow us to compute an ....

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial dierential equations by mixed nite element methods. Numer. Math., 65 (1993) 95-108.

No context found.

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang. A parallel iterative procedure applicable to the approximate solution of second order partial dierential equations by mixed nite element methods. Numer. Math., 65 (1993) 95-108.

No context found.

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods, Numer. Math., 65(1993), pp. 95-108.

....parameter is given. As an aside in his thesis on the application of this technique to the more difficult Helmholtz problem, Despr es [1] extended Lions s proof to a more general partition but again without an estimate of an explicit rate of convergence. Douglas, Paes Leme, Roberts, and Wang [2] did obtain convergence rates for fixed fi for a mixed finite element approximation of (1) under a number of different hypotheses on the coefficients and the partition. 2 The paper is organized as follows. In x2, we consider the differential model problem in the plane and indicate how its ....

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang. A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods. Numer. Math., 65 (1993) 95--108.

....The use of Robin boundary conditions as interfacial transmission conditions in nonoverlapping domain decomposition iterative procedures was introduced by P. L. Lions [7] various aspects of such methods have been discussed by Lions, Despr es [3] and Douglas, Paes Leme, Roberts, and Wang [5], among others. In all of these references, the weighting fi of the normal derivatives in the transmission condition fi u j j u j = Gammafi u k k u k on the interface Gamma jk between subdomains Omega j and Omega k (as looked at from Omega j ) has been independent of the ....

....j u j = fiq k Delta k u k ; x 2 Gamma jk ; j; k = 1; 2; k 6= j; 2.6) are equivalent to (2.5) and a nonoverlapping domain decomposition procedure can be based on (2.6) as first described by P. L. Lions [7] We want to apply a mixed finite element procedure to (2.4) and (2. 5) As in [5], the consistency conditions (2.5) or (2.6) will lead us to hybridize the mixed method. Let V j Theta W j Theta jk = V( Omega j ) Theta W( Omega j ) Theta ( Gamma jk ) be a mixed finite element space over f Omega j g, where V j is the space of (flux) vectors on Omega j , W j the space ....

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang. A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods. Numer. Math., 65 (1993) 95--108.

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