Marcus V. Sarkis. Schwarz Preconditioners for Elliptic Problems with Discontinuous Coecients Using Conforming and Non-Conforming Elements. PhD thesis, Courant Institute, New York University, September 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... the fractional order Sobolev spaces, 0 1) de ned by the completion of C in the following norm, jjujj L juj H ) juj H = Z ju(x) u(y)j jx yj d 2 dx dy: A more detail introduction to the important tools used in domain decomposition theory can be found in [41, 6, 44]. 1.2.1 Trace Theorems For a continuous function u , the trace of u can be simply de ned by restricting u to The trace theorems extend this de nition to more general functions; see [1] for the general theory. is a Lipschitz domain and u 2 H , 1=2 s 1, then, 0 u = u j ( ....

....problem with Dirichlet boundary conditions on parts of boundary we need to consider a Sobolev space H ( fu 2 H ju j = 0g. The Poincar e Friedrichs Inequality gives an equivalence of norms on this space. The idea of its proof can be found in [35] and we can also nd a proof in [41]. Theorem 1.4 (Poincar e Friedrichs Inequality) Let with positive measure. Then, H 0 u d ) is a constant that is invariant under dilation of and . In Chapter 6, we will use the classical Friedrichs Inequality, Theorem 1.5 (Friedrichs Inequality) ....

Marcus V. Sarkis. Schwarz Preconditioners for Elliptic Problems with Discontinuous Coecients Using Conforming and Non-Conforming Elements. PhD thesis, Courant Institute, New York University, September 1994.

....The sets of nodes on ; i ; and are denoted by h ; i;h ; and h ; respectively. As in previous work on Neumann Neumann and FETI algorithms, a crucial role is played by the weighted counting functions i ; which are associated with the individual subdomain boundaries i ; cf. [5, 8, 14, 19]. In this paper they will be used in the de nition of certain diagonal scaling matrices. These functions are de ned, for 2 [1=2; 1) and for x 2 h [ h , by a sum of contributions from i , and its relevant next neighbors i (x) 8 : X j2Nx j (x) x 2 i;h j;h ....

Marcus V. Sarkis. Schwarz Preconditioners for Elliptic Problems with Discontinuous Coe- cients Using Conforming and Non-Conforming Elements. PhD thesis, Courant Institute, New York University, September 1994.

.... 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 ....

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Marcus V. Sarkis. Schwarz Preconditioners for Elliptic Problems with Discontinuous Coe#- cients Using Conforming and Non-Conforming Elements. PhD thesis, Courant Institute, New York University, September 1994. TR671, Department of Computer Science, New York University, URL: file://cs.nyu.edu/pub/tech-reports/tr671.ps.Z.

....of the H(div ; Omega Gamma and H(curl ; Omega Gamma cases for three dimensional problems. Among them are [9, 14, 23] on two level overlapping methods, 3, 13] on multilevel methods, and [1] which is a study of an iterative substructuring method in H(curl ; Omega Gamma1 We also mention [20, 21] which report on a study of a class of two and multi level methods for mixed approximations of Poisson s equation. The present work is a continuation of our recent work in two dimensions; see [5] See also [2, 7, 12, 15, 16] and the references therein, for some Schwarz methods for problems in ....

Marcus V. Sarkis. Schwarz Preconditioners for Elliptic Problems with Discontinuous Coefficients Using Conforming and Non-Conforming Elements. PhD thesis, Courant Institute, New York University, September 1994.

.... 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. Schwarz preconditioners for elliptic problems with discontinuous coefficients using conforming and non-conforming elements. PhD thesis, Courant Institute of the Mathematical Sciences, New York University, 1994.

.... Schwarz preconditioners for H(div ; and H(curl ; were initially developed for two dimensions, in [3] and then extended to three dimensions, in [32, 19] Multigrid and multilevel methods were considered in [3, 2, 18, 17, 4, 19] and iterative substructuring methods in [1, 34, 36] We also mention [16, 13, 6, 23, 24, 8, 29, 30], which report on a study of a class of two and multi level methods for mixed approximations of Poisson s equation. An important element in the de nition of a Neumann Neumann method is a set of scaling functions de ned on the boundaries of the substructures, which involve the values of the ....

....both coecients a and B in (3) This is due to the particular divergence free extension employed in the proof of Lemma 6.2; see Lemma 5.2. Because of the nature of the degrees of freedom in X 0;h , our scaling functions are particularly simple, compared to those for problems in H 1( 7 Following [28, 29], our family of scaling functions depend on a parameter 1=2: 25) Let T be a substructure. We de ne a piecewise constant function T 2 S h ( T ) by T j f P D D T ; f 2 F h ; f T ; 26) where D and T are the largest eigenvalues of the coecient matrices BD and B T , ....

Marcus V. Sarkis. Schwarz Preconditioners for Elliptic Problems with Discontinuous Coe- 24 cients Using Conforming and Non-Conforming Elements. PhD thesis, Courant Institute, New York University, September 1994.

....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. Schwarz preconditioners for elliptic problems with discontinuous coefficients using conforming and non-conforming elements. PhD thesis, Courant Institute of the Mathematical Sciences, New York University, 1994.

....have been only relatively few studies of the H(div ; and H(curl ; cases for three dimensional problems. Among them are [9, 14, 23] on two level overlapping methods, 3, 13] on multilevel methods, and [1] which is a study of an iterative substructuring method in H(curl ; 1 We also mention [20, 21] which report on a study of a class of two and multi level methods for mixed approximations of Poisson s equation. The present work is a continuation of our recent work in two dimensions; see [5] See also [2, 7, 12, 15, 16] and the references therein, for some Schwarz methods for problems in ....

Marcus V. Sarkis. Schwarz Preconditioners for Elliptic Problems with Discontinuous CoeĈ- cients Using Conforming and Non-Conforming Elements. PhD thesis, Courant Institute, New York University, September 1994.

.... zero and f depends on the solution at the previous steps, as well as on the right hand side of (1) In the last few years, a considerable effort has been devoted to the study of Schwarz methods for the solution of linear systems arising from non conforming finite element problems; see [20] 21] [26], 2] 14] 12] 15] 19] 28] Our analysis of overlapping methods has been inspired by [2] where a Schwarz method for a non conforming finite element problem in 2D is studied; their result is valid for 2D Maxwell s equations; see [28] In addition, we will also use the technical tools and ....

Marcus V. Sarkis. Schwarz Preconditioners for Elliptic Problems with Discontinuous Coefficients Using Conforming and Non-Conforming Elements. PhD thesis, Courant Institute, New York University, September 1994.

....on decomposing the domain into subdomains of size d and involve the solution of related problems on the subdomains and lower order coupling systems on the subdomain boundaries. The best condition number for the preconditioned system is shown to be on the order of (1 ln (d=h) ff , ff = 1; 2, [8, 15, 25]) where h is the mesh size parameter. Another approaches consist in developing the bordering method [21, 22] or capacitance matrix method [5, 12, 14] The main idea is to use well known techniques to solve or precondition the problems in the subdomains. For the problem at the interfaces a ....

M. Sarkis, Schwarz Preconditioners for Elliptic Problems with Discontinuous Coefficients Using Conforming and Non-conforming Elements, PhD thesis, Courant Institute of Mathematical Sciences, New York University, New York, September 1994.

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Marcus V. Sarkis. Schwarz Preconditioners for Elliptic Problems with Discontinuous Coefficients Using Conforming and Non-Conforming Elements. PhD thesis, Courant Institute, New York University, September 1994.

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