M. Sarkis, Two-Level Schwarz Methods for Nonconforming Finite Elements and Discontinuous Coefficients. TR 629, Courant Institute of Mathematical Sciences, New York, 1993  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that can be obtained by taking advantage of techniques due to Cowsar [13] in the framework of domain decomposition methods (for 1 Math. Institute, University of Augsburg, D 86135 Augsburg, Germany. E mail: hoppe math.uni augsburg.de, wohlmuth math.uni augsburg.de similar ideas cf. also Sarkis [29]) The multilevel iterative solution process will go hand in hand with an adaptive grid refinement technique relying on the element oriented error estimator developed by the authors in [18] We consider the following boundary value problem for a linear second order elliptic differential operator ....

M. Sarkis, Two-Level Schwarz Methods for Nonconforming Finite Elements and Discontinuous Coefficients. TR 629, Courant Institute of Mathematical Sciences, New York, 1993

.... introduce a new one parameter family of FETI preconditioners and prove a bound on the rate of convergence which is independent of possible jumps of the coe#cients of an elliptic model problem previously considered in the theory of Neumann Neumann and other iterative substructuring algorithms; see [11, 8, 23, 30, 31]. In fact, we have found it possible to reduce the analytic core of the theory for the new class of FETI methods to a variant of an estimate # SCAI Institute for Algorithms and Scientific Computing, GMD German National Research Center for Information Technology, Schloss Birlinghoven, ....

....algorithmic idea used in the best of the Neumann Neumann methods. A proof of one of the two spectral bounds that are required then becomes just as elementary as for the Neumann Neumann case. We note that our family of scalings of the preconditioner was apparently first introduced by Sarkis [30, 31]; see also [7] The other scaling a#ects the choice of the projection which is used in each step of the FETI iteration, whether preconditioned or not. We will show that, for a certain choice of the two scalings, our preconditioner is the same as one recently tested successfully for very di#cult ....

Marcus V. Sarkis. Two-level Schwarz methods for nonconforming finite elements and discontinuous coe#cients. In N. Duane Melson, Thomas A. Manteu#el, and Steve F. McCormick, editors, Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, Volume 2, number 3224, pages 543--566, Hampton VA, 1993. NASA.

....methods, based on multigrid, multilevel or domain decomposition, for solving or preconditioning the reduced problem; c) recovery of the solution u(x) and the velocity q from the Lagrange multipliers that were already found. The recent progress in each of the steps described above (see, e.g. [20, 18, 10]) gives us an indication that the mixed finite element method can be used as an accurate and efficient tool for solving general elliptic problems of second order in domains with complicated geometry. The goal of this paper is to construct, study and implement efficient preconditioners for the ....

M. Sarkis, Two-level Schwarz methods for nonconforming finite elements and discontinious coefficients, Preprint, 1993.

....methods, based on multigrid, multilevel or domain decomposition, for solving or preconditioning the reduced problem; c) recovery of the solution u(x) and the velocity q from the Lagrange multipliers that were already found. The recent progress in each of the steps described above (see, e.g. [20, 18, 10]) gives us an indication that the mixed finite element method can be used as an accurate and efficient tool for solving general elliptic problems of second order in domains with complicated geometry. The goal of this paper is to construct, study and implement efficient preconditioners for the ....

M. Sarkis, Two-level Schwarz methods for nonconforming finite elements and discontinious coefficients, Preprint, 1993.

.... The matrix A is a good preconditioner for A, i.e. 9a 0 ; a 1 0 a 2 0 u t Au u t Au a 2 1 u t Au 8u 2 X: 21) The constants a 0 ; a 1 should preferably be independent of the discretization parameters but there are also interesting cases with a polylogarithmic dependence on H=h; see [17,34,35]. The parameter H represents the diameter of a subdomain in a domain decomposition method. Multigrid and domain decomposition methods are examples of preconditioners that meet these requirements. They have also been successfully implemented on parallel machines; see e.g. for details ....

Marcus Sarkis. Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients. In N. Duane Melson, Thomas A. Manteuffel, and Steve F. McCormick, editors, Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, Volume 2, number 3224, pages 543--566, Hampton VA, 1993. NASA.

....are given by domain decomposition and multigrid methods or, more generally, by Schwarz methods; see Section 2.2. The ellipticity constants should preferably be independent of the discretization parameters but there are also interesting cases with a polylogarithmic dependence on H=h; see [37, 72, 73]. The parameter H represents the diameter of a subdomain in a domain decomposition method. 4.2.1 A Condition Number Estimate In view of Theorem 2.1, our goal is to give an estimate of the condition number ( B Gamma1 A) ae( B Gamma1 A)ae( B Gamma1 A) Gamma1 ) Since B ....

Marcus Sarkis. Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients. In N. Duane Melson, Thomas A. Manteuffel, and Steve F. McCormick, editors, Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, Volume 2, number 3224, pages 543--566, Hampton VA, 1993. NASA.

....i is a local measure of the stiffness of subdomain Omega i (for example an average Young modulus on Omega i ) and ae s = P P l 2 Omega j ae s l is the sum of ae s on all subdomains Omega j containing P l . In our computations, we choose s = 1; the choice s = 1=2 was used in [16] In [28, 29, 36], it was realized (though not for plate problems) that any s 1=2 results in the condition number bounded independently of the values ae i , i.e. independently of coefficient jumps between subdomains. We will see that the proposed preconditioner is efficient for plate problems if all elements of ....

M. Sarkis, Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients, in Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, Volume 2, N. D. Melson, T. A. Manteuffel, and S. F. McCormick, eds., no. 3224, Hampton VA, 1993, NASA, pp. 543--566.

.... work, in which such a framework is developed and used, see Bramble, Pasciak, Wang, and Xu [8] Cai [10] Cai and Widlund [12] 13] Dryja and Widlund [23] 24] 26] 27] 28] 29] Lions [35] Mathew [42] 43] Nepomnyaschikh [45] Pavarino [47] 48] Pavarino and Widlund [49] Sarkis [51], Smith [54] 55] 56] 57] Widlund [61] Xu [63] and Zhang [64] 66] In Section 2, we will demonstrate that rapid convergence of the iterative methods occurs if and only if all u 2 V can be decomposed into components in V i ; i.e. u = P i u i ; u i 2 V i ; in such a way that P b i (u i ....

M. Sarkis, Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients, Tech. Rep. 629, Courant Institute of Mathematical Sciences, Department of Computer Science, March 1993.

.... work in which such a framework is developed and used, see Bramble, Pasciak, Wang, and Xu [8] Cai [10] Cai and Widlund [12] 13] Dryja and Widlund [24] 25] 27] 28] 29] 30] Lions [36] Mathew [43] 44] Nepomnyaschikh [46] Pavarino [48] 49] Pavarino and Widlund [50] 51] Sarkis [53], Smith [56] 57] 58] 59] Widlund [63] Xu [65] and Zhang [66] 68] In Section 2, we will demonstrate that rapid convergence of the iterative methods occurs if and only if all u 2 V can be decomposed into components in V i ; i.e. u = P i u i ; u i 2 V i ; in such a way that P b i ....

M. Sarkis, Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients, Tech. Rep. 629, Courant Institute of Mathematical Sciences, Department of Computer Science, March 1993, also in the Proc. of the Sixth Copper Mountain Conf. on Multigrid Methods, pp. 543-566, NASA Conference Publication 3224, Part II, 1993.

....coarse mesh space, we prove that uniformly bounded decomposition of V h can be obtained. The use of non nested spaces in multigrid methods has been studied in [2] and in domain decomposition methods in [9] Other Schwarz algorithms that use nonconforming elements can be found in [20]. We begin by introducing several notations. Let Omega Hc = f i g be a quasi uniform triangulation of Omega and i one of the triangles whose diameter is of order H c . Omega Hc will be referred to as the coarse grid. Here H c is the maximum diameter of this coarse triangulation. We assume, ....

M. Sarkis, Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients, TR-580, Courant Institute, New York Univ., 1993.

....V corresponds to the k th degree of freedom of V i , and D i v(P l ) 0; otherwise. Here, s 1=2, and ae s = P P l 2 Omega j ae s j is the sum of ae s j for all subdomains Omega j containing the node P l . In our computations, we choose s = 1; the choice s = 1=2 was used in [16] In [29, 28, 37], it was established for second order elliptic problems that any s 1=2 results in a condition number bounded independently of the values ae i , i.e. independently of the coefficient jumps between subdomains. We will see that the proposed preconditioner is efficient for plate and shell problems ....

M. Sarkis, Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients, in Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, Volume 2, N. D. Melson, T. A. Manteuffel, and S. F. McCormick, eds., no. 3224, Hampton VA, 1993, NASA, pp. 543--566.

.... Omega ; 3.15c) 8 CHEN, EWING, AND LAZAROV where N 1 and N 2 are the number of the adjacent midpoints (x 0 j ; y 0 j ) and (x 00 j ; y 00 j ) to (x i ; y i ) of the edges in E k Gamma1 and the edges on Omega of the elements in E k Gamma1 , respectively. Alternatively, following [36], I k k Gamma1 : N k Gamma1 N k can be equivalently defined by I k k Gamma1 v(x i ; y i ) v(x i ; y i ) i = 1; 2; 3; 3.16a) I k k Gamma1 v(x i ; y i ) 1 N 1 X (x i ;y i )2K j vj K j (x i ; y i ) if (x i ; y i ) 2 Omega ; 3.16b) I k k Gamma1 v(x i ; y i ) 1 N 2 X j ....

....Then there is a constant C independent of H and ffi such that for v 2 N i h (3. 25) jjvjj 2 L 2 ( Gamma ffi;i ) C ffi 2 i (1 H=ffi)jvj 2h; Omega i (H ffi ) Gamma1 jjvjj 2 L 2 ( Omega i ) j : DOMAIN DECOMPOSITION ALGORITHMS FOR MIXED METHODS 11 Lemma 8 was proven in [36]. Lemma 9 was shown for any element in H 1 ( Omega i ) in [21] Its proof is local and can be easily extended to the elements in N i h . Proof of Theorem 4. Let N 0 h be given in (3.17) The spectrum of Pi can be bounded as before. It thus suffices to prove the assumption (A1) Let f i g ....

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M. Sarkis, Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients, Dept. of Computer Science TR 93-629, Courant Institute, 1993.

....The method of Dryja and Widlund [32] uses the same coarse space W as BDD and weight exponent t = 1=2 in (2.25) but instead of using the local Schur complement matrices S (i) in (2.18) it uses S (i) c i M (i) with M (i) positive definite. This avoids solving singular problems. Sarkis [79] obtained an estimate for a method similar to BDD for nonconforming elements with any t 1=2. As the main purpose of this brief section is to give a few references to other material deserving attention but not covered in this thesis, we note that domain decomposition methods have been developed ....

M. Sarkis, Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients, in Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, Volume 2, N. D. Melson, T. A. Manteuffel, and S. F. McCormick, eds., no. 3224, Hampton VA, 1993, NASA, pp. 543--566.

....of spaces) one may use a two level method to switch to a conforming reference discretization for the nonconforming discretization on V J which incorporates known (multilevel) preconditioners for the conforming relative . The same idea is used in [6] in a domain decomposition framework, see also [9,18]. As a rule, this approach (as well as the use of variable V cycle preconditioners) gives asymptotical optimal algorithms, in the sense, that the condition numbers remain bounded for J 1, and the work per iteration is proportional to the number of unknowns. 3 Discussion and examples 3.1 ....

M. Sarkis, Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients, in: N.D.Melson et al., eds., Proceedings Sixth Copper Mountain Conference on Multigrid Methods (NASA Conference Publications, vol. 3224, 1993) 543--565.

....variations only across the interfaces of these substructures. We then design methods with several coarse spaces, sometimes known as exotic coarse spaces; cf. Widlund [25] Some are new and others have previously been discussed; see Dryja, Smith and Widlund [10] Dryja and Widlund [15] and Sarkis [20]. One of our main results is that the condition number of the resulting systems can be estimated from above by C (1 ) 2 with C independent of the jumps of coefficients, of the number of substructures, and also of . For multiplicative variants such as the V cycle multigrid, the rate of ....

....the standard piecewise linear functions can be used as the coarse space and optimal multilevel algorithms are obtained. In Section 7, we introduce approximate discrete harmonic extensions and define new coarses spaces by modifying the previously known exotic coarse spaces; see Sarkis [20] for a case of nonconforming elements. Using 2 these extensions, we can avoid solving a local Dirichlet problem for each substructure when using exotic coarse spaces. We show that the converge rate estimate of our new iterative methods, with approximate discrete harmonic extensions, are ....

[Article contains additional citation context not shown here]

Marcus Sarkis. Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients. In N. Duane Melson, Thomas A. Manteuffel, and Steve F. McCormick, editors, Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, Volume 2, number 3224, pages 543--566, Hampton VA, 1993. NASA.

....both two and three dimensions. We define and analyze new coarse spaces and obtain condition numbers with just one log factor. A short version of this paper was entered into Copper Mountain student competition in mid December 1992. The present paper is a slightly modification of a technical report [8]. DIFFERENTIAL AND FINITE ELEMENT MODEL PROBLEMS To simplify the presentation, we assume that Omega is an open, bounded, polygonal region of diameter 1 in the plane, with boundary Omega : In a separate section, we extend all our results to the three dimensional case. We introduce a partition ....

Marcus Sarkis. Two-level Schwarz methods for nonconforming finite elements and discontinuous coefficients. Technical Report 629, Department of Computer Science, Courant Institute, March 1993.

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