M. A. Casarin. Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids. PhD thesis, Courant Institute of Mathematical Sciences, March 1996. Tech. Rep. 717, Department of Computer Science, Courant Institute.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....There are extensive applications of domain decomposition methods to saddle point problems, especially for incompressible Stokes problems. Previous domain decomposition methods for incompressible Stokes equations have been based on primal iterative substructuring methods, cf. 1] 7] 11] [12], 13] 35] 59] 60] 61] 64] 66] 69] 76] on overlapping Schwarz methods, cf. 34] 36] 48] 70] and on block preconditioners, cf. 47] 49] A discussion of overlapping Schwarz methods and iterative substructuring methods for solving incompressible Stokes problems is given in ....

....which constrain the continuity of velocity on the subdomain interface. Since no coarse level solver is implemented in each iteration step, the convergence of CG depends on the number of subdomains. Some other domain decomposition methods for incompressible Stokes equations can be found in [1] [12], 13] 34] 35] 36] 59] 60] 61] 66] 69] 70] Other approaches for the iterative solution of saddle point problems are Uzawa s algorithm, cf. 6] 22] 55] multigrid methods, cf. 4] 5] 9] 79] 83] preconditioned conjugate gradient methods for a positive de nite ....

Mario A. Casarin. Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids. PhD thesis, Courant Institute of Mathematical Sciences, March 1996. Tech. Rep. 717, Department of Computer Science, Courant Institute.

....obtained by spectral and p version finite elements are less sparse and more ill conditioned than those obtained with h version finite elements. The construction and analysis of efficient preconditioned iterative methods is therefore more challenging. We refer to Pavarino and Widlund [31] Casarin [14] and to the references therein for an overview of recent results based on domain decomposition techniques for elliptic scalar problems. In the context of spectral elements for Stokes and Navier Stokes problems, iterative methods have been studied in Maday, Meiron, Patera and Rnquist [25] Maday, ....

....C = C. Interesting choices for A are given by h version finite element discretizations on the GLL mesh or by substructuring domain decomposition methods, where a 0 and a 1 have a polylogarithmic dependence on the spectral degree n (for the scalar case, see Pavarino and Widlund [32] and Casarin [14]) Since the resulting preconditioned system is symmetric, we can use the Preconditioned Conjugate Residual Method (PCR) see Ashby, Manteuffel and Saylor [2] and Hackbusch [21] See Elman [16] for a short description of the ORTHMIN version of PCR for symmetric indefinite systems. In his thesis ....

M. Casarin, Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids, PhD thesis, Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, 1996.

....and FETI methods has been considered recently by Klawonn and Widlund [20] There is a considerable literature on domain decomposition methods for incompressible Stokes equations. Iterative substructuring methods have been studied by Ainsworth and Sherwin [2] Bramble and Pasciak [7] Casarin [9], Fischer and R nquist [15] Le Tallec and Patra [22] Marini and Quarteroni [26] Pasciak [27] Pavarino and Widlund [29] Quarteroni [31] and R nquist [33] Overlapping Schwarz methods have been considered by Fischer [13] Fischer, Miller, and Tufo [14] Gervasio [16] Klawonn and Pavarino ....

.... q) 0 8q 2 U i u = u on (16) The following comparison of the energy of the discrete saddle point harmonic extension operator and the discrete harmonic extensions H of each displacement component separately is a generalization of the analogous comparison in the Stokes case (see [7] 22] [9, 10], 2] Lemma 3.1. Given u 2 e V j , let H(u ) be its componentwise discrete harmonic extension and let SH ; u ) be its discrete saddle point harmonic extension. Then, 8u 2 V such that u ker(a i ) krHu k i ) C where is the inf sup ....

Mario A. Casarin. Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids. PhD thesis, Dept. of Mathematics, Courant Institute of Mathematical Sciences, New York University, March 1996. Tech. Rep. 717, Department of Computer Science, Courant Institute.

.... 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 ( ....

Mario A. Casarin. Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids. PhD thesis, 136 Courant Institute of Mathematical Sciences, March 1996. Tech. Rep. 717, Department of Computer Science, Courant Institute.

....problems is here extended to the threedimensional case. The update of the geometry is decoupled from the solution of the steady Navier Stokes equations; this paper emphasizes the latter step. In terms of providing insight into why the method in [Rn96] works as well as it does, we refer to [Cas96]. In [Cas96] other solution approaches are also proposed for the steady Navier Stokes equations. However, no numerical results exist yet for these alternative approaches. Ninth International Conference on Domain Decomposition Methods Editor Petter E. Bjrstad, Magne S. Espedal and David E. Keyes ....

....is here extended to the threedimensional case. The update of the geometry is decoupled from the solution of the steady Navier Stokes equations; this paper emphasizes the latter step. In terms of providing insight into why the method in [Rn96] works as well as it does, we refer to [Cas96] In [Cas96] other solution approaches are also proposed for the steady Navier Stokes equations. However, no numerical results exist yet for these alternative approaches. Ninth International Conference on Domain Decomposition Methods Editor Petter E. Bjrstad, Magne S. Espedal and David E. Keyes c fl1998 ....

Casarin M. (1996) Schwarz preconditioners for spectral and mortar finite element methods with applications to incompressible fluids. PhD thesis, New York University.

....similar reasons, we also exclude configurations in which the endpoint of one PC interface connects to the interior of another, as shown in Fig. 4c. Much work has been done on nonconforming spectral element methods, starting with the early work of Mavriplis [26] Anagnostou et al. 1] and others [3, 18, 9]. Most of these have employed mortar elements that increase flexibility through the use of L 2 projection operators to enforce weak continuity at the nonconforming interface. In particular, the vertex free mortar spaces of Ben Belgacem and Maday [3] alleviate the restriction of Fig. 4c. The ....

M. Casarin, "Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids," Ph.D. Thesis, Technical Report 717, Dept. of Computer Science, Courant Institute (1996). 18

....element discretizations of the linear elasticity and Stokes systems in three dimensions. For other approaches to the iterative solution of spectral element methods for Stokes and Navier Stokes problems, see Maday, Patera and Rnquist [MPR92] Fischer and Rnquist [FR94] Rnquist [Rn96] Casarin [Cas96] and the references therein. For p version finite element preconditioners for elasticity, see Mandel [Man96] Let Omega ae R 3 be a polyhedral domain and Gamma 0 a subset of its boundary. Let V be the Sobolev space V = fv 2 H 1( Omega Gamma 3 : vj Gamma 0 = 0g. The linear elasticity problem ....

.... W n : Interesting choices for A are given by h version finite element discretizations on the GLL mesh or by substructuring domain decomposition methods, where a 0 and a 1 have a polylogarithmic dependence on the spectral degree n (for the scalar case, see Pavarino and Widlund [PW96] and Casarin [Cas96]) Since the resulting preconditioned system is symmetric, we can use the Preconditioned Conjugate Residual Method (PCR) see Hackbusch [Hac94] Combining Klawonn s result ( Kla96] pp. 46 47) and Theorem 1, we obtain the following convergence result. Theorem 2. If K is the stiffness matrix of ....

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Casarin M. (1996) Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids. PhD thesis, Dept. of Math., Courant Institute, New York University.

....by Deville, Mund, and coworkers, e.g. 9, 10] The combination of spectral methods, finite element preconditioning, and additive Schwarz methods has been investigated by Pahl [34] Pavarino and Widlund [37] and Casarin [5] for the case of the discrete Laplacian. Rnquist [40] and Casarin [6] have studied iterative substructuring methods for spectral element solution of the fully coupled steady Navier Stokes equations. Rnquist also proposed a block Jacobi deflation based scheme applied to the consistent 2 (a) b) present Koumoutsakos Leonard CD time (c) Figure 1: a) ....

....of a regularly structured array of subdomains. Thus, E has no natural interface structure, or separator, such as commonly found in operators derived from finite element bases of compact support. Consequently, E cannot be readily treated by substructuring, or Schur complement, approaches as in [6, 37]. In addition, there are no boundary conditions associated directly with the pressure, as boundary conditions for the Stokes problem are applied in the velocity space. Despite the L 2 approximation space used for the pressure it is nonetheless clear that E is related to a Laplacian on ....

M. Casarin "Schwarz preconditioners for spectral and mortar finite element methods with applications to incompressible fluids," Ph.D. Thesis, Courant Institute of Math. Sci., NYU, 1996.

....incompressible Stokes equations. Previous work on domain decomposition methods for incompressible Stokes equations has been based on other iterative substructuring methods, see Bramble and Pasciak [6] Pasciak [28] Quarteroni [31] Marini and Quarteroni [27] Fischer and R nquist [12] Casarin [8], R nquist [33] Le Tallec and Patra [23] Pavarino and Widlund [30] and Ainsworth and Sherwin [2] on overlapping Schwarz methods, see Gervasio [14] Fischer [11] Fischer, Miller, and Tufo [13] Klawonn and Pavarino [19] and R nquist [34] and on block preconditioners, see Klawonn [18, 17] ....

.... u; v) b(v; p) 0 8v 2 V i b( u; q) 0 8q 2 U i u = u on i : The following comparison of the energy of the discrete Stokes extensions SH and the discrete harmonic extensions H of each velocity component can be found in [15] and [6] for nite element discretizations and in [23] and [8] for spectral element discretizations. We note that the corresponding local bounds, for individual subdomains are equally valid and that the upper bound has an elementary proof. Lemma 1 c a(SHu ; SHu ) a(Hu ; Hu ) a(SHu ; SHu ) 8u 2 V ; where is the inf sup constant of the chosen mixed ....

M. A. Casarin, Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids, PhD thesis, Dept. of Mathematics, Courant Institute of Mathematical Sciences, New York University, March 1996.

....methods has been investigated by Pahl [28] Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439, USA. y Radex Inc. Bedford, MA 01730, USA. 1 2 P.F. FISCHER, N.I. MILLER, AND H.M. TUFO Pavarino and Widlund [30] and Casarin [5] Rnquist [33] and Casarin [6] have studied iterative substructuring methods for spectral element solution of the fully coupled steady Navier Stokes equations. Rnquist also proposed a block Jacobi deflation based scheme applied to the consistent Poisson operator governing the pressure for the unsteady case [15, 32] The ....

M. Casarin, "Schwarz preconditioners for spectral and mortar finite element methods with applications to incompressible fluids", PhD. Thesis, Courant Institute of Math. Sci., NYU (1996).

....domain decomposition methods for non matching grids. The active research by the scientific computation community in this field is motivated by its flexibility and great potential for large scale parallel computation (see, e.g. 3] A good description of the mortar element method can be found in [2, 6, 7, 12]. The nonconforming finite element mortar method has been studied in [7] where optimal order convergence in H 1 norm was demonstrated. Three dimensional mortar finite element analysis has been given in [5] Nonmortar mixed finite element approximations for second order elliptic problems have ....

M. A. Casarin, Schwarz preconditioners for spectral and mortar finite element methods with applications to incompressible fluids, TR 1996, Dept of Computer Sciences, New York University.

....these methods, improved accuracy is achieved by increasing the spectral degree aas well as the number of elements. We note that iterative solvers for a variety of higher order methods have been developed by Mandel [24] Katz and Hu [18] and Guo [16] see also the theses of Pavarino [28] Casarin [9], and Bica [4] In our previous work [29, 31] we considered the scalar case and iterative substructuring methods with a wire basket based coarse space. Each spectral element is then the affine image of a reference cube considered as a subdomain of the domain decomposition method. The wire basket ....

....the elements) We proved, and verified numerically that the convergence rate of this method is independent of the number of elements and the jumps in the coefficients of the elliptic operator, and that it depends only weakly on the spectral degree. An alternative proof was later given by Casarin [9]. This type of wire basket preconditioner was originally proposed for h version finite elements by Smith [32, 33, 34] see also Bramble, Pasciak and Schatz [5] for earlier related work. Other iterative substructuring methods that have been successfully applied to elasticity problems and h version ....

M. A. Casarin, Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids, PhD thesis, Dept. of Mathematics, Courant Institute of Mathematical Sciences, New York University, March 1996.

....for an introduction to spectral methods. Iterative substructuring methods for spectral and hp discretizations of Stokes and Navier Stokes problems can be found in Quarteroni [36] Fischer and Rnquist [20] Maday, Meiron, Patera, and Rnquist [26] Rnquist [37] Le Tallec and Patra [25] and Casarin [14]. For h version finite elements, iterative substructuring methods for Stokes problems can be found in Bramble and Pasciak [6] while multigrid methods for mixed linear elasticity have been studied by Brenner [7, 8, 9] In this paper, we extend the iterative substructuring methods, previously ....

.... Stokes extension satisfies the minimization property s n (S n u; S n u) min vj Gamma =u s n (v; v) 8v 2 fv 2 V n : b n (v; q) 0 8q 2 N X i=1 U n i g: The following comparison of the energy of the discrete Stokes and harmonic extensions can be found in [21] 6] 25] and [14]. Lemma 5.1. cfi n s n (S n u; S n u) s n (H n u; H n u) s n (S n u; S n u) 8u 2 V n ( Gamma) 5.3. The discrete mixed elastic extension. We can also extend a piecewise polynomial from Gamma to the interior of each element by solving an incompressible linear elasticity problem ....

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M. A. Casarin, Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids, PhD thesis, Dept. of Mathematics, Courant Institute of Mathematical Sciences, New York University, March 1996.

.... For positive definite systems, see, e.g. Mandel [27, 26] Le Tallec [22] and Farhat and Roux [16] and for saddle point problems, see Bramble and Pasciak [5] Quarteroni [34] Fischer and Rnquist [17] Maday, Meiron, Patera, and Rnquist [24] Rnquist [35] Le Tallec and Patra [23] and Casarin [10]. We also note that alternative iterative methods have been considered for saddle point problems, such as Uzawa s algorithm, multigrid methods, block diagonal and block triangular preconditioners; see, e.g. Elman [13, 14] Brenner [6] Klawonn [20] and the references therein. The rest of the ....

.... Stokes extension satisfies the minimization property s n (S n u; S n u) min vj Gamma =u s n (v; v) 8v 2 fv 2 V n : b n (v; q) 0 8q 2 N X i=1 U n i g: The following comparison of the energy of the discrete Stokes and harmonic extensions can be found in [18] 5] 23] and [10]. Lemma 4.1. cfi n s n (S n u; S n u) s n (H n u; H n u) s n (S n u; S n u) 8u 2 V n Gamma : 4.4. The discrete mixed elastic extension. We can also extend any element of V n Gamma to the interior of each element by solving an incompressible linear mixed form elasticity ....

[Article contains additional citation context not shown here]

M. A. Casarin, Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids, PhD thesis, Dept. of Mathematics, Courant Institute of Mathematical Sciences, New York University, March 1996.

....domain decomposition methods for nonmatching grids. The active research by the scientific computation community in this field is motivated by its flexibility and great potential for large scale parallel computation (see, e.g. BM94] A good description of the mortar element method can be found in [Bel97, BDM90, BMP94, Cas]. The nonconforming finite element mortar method has been studied in [BMP94] where optimal order convergence in H 1 norm was demonstrated. Three dimensional mortar finite element analysis has been given in [BM97] Non mortar mixed finite element approximations for second order elliptic 1 ....

Casarin M.Schwarz preconditioners for spectral and mortar finite element methods with applications to incompressible fluids. Technical report, Department of Computer Sciences.

....associated with the overlapping subdomains. The work in [KPar] was done in the context of stable, mixed, linear finite elements. In the context of high order spectral element methods, primarily iterative substructuring methods have been considered for the global, coupled Stokes system in the past [Rn96, Rn98, Cas96, PW97]; the techniques reported in [Rn96, Rn98] have also been extended to solve the steady Navier Stokes equations in three dimensions. Recently, however, the methodology presented in [KPar] has also been extended to spectral elements; see [Pavar] In this paper, we consider a new overlapping, ....

....components of f are the nodal forces. Boldface symbols are associated with multi dimensional quantities, while non boldface symbols are associated with scalar quantities. Solution methods There exist many iterative algorithms for solving the indefinite saddle point problem (1) see for example [MMPR93, Elm96, Cas96, PW97, Pav97, Rn96]. In this section, we will focus our attention on a few iterative methods. In the next section, numerical results will be presented, and the methods will be compared. Method U: The Uzawa algorithm A classical solution method, the Uzawa algorithm, consists of first decoupling the Stokes saddle ....

Casarin M. (1996) Schwarz preconditioners for spectral and mortar finite element methods with applications to incompressible fluids. PhD thesis, New York University, Courant Institute of Mathematical Sciences.

....elliptic equations, Schwarz preconditioner AMS(MOS) subject classifications. 65N30, 65F10 1. Introduction. The mortar element method was first developed for the purpose of coupling di#erent discretizations in di#erent nonoverlapping subdomains. Several studies have been carried out; see e.g. [1, 2, 3, 4, 5, 6, 7, 11, 12, 15, 16, 22, 25, 29, 30]. In this paper, we consider the case of overlapping subdomains. We provide an optimal error analysis for the two subdomain case, and also spectral bound estimations for the Schwarz preconditioned systems. The main advantage of non matching grid methods is that highly structured local grids and ....

M. Casarin, Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids, PhD thesis, Courant Institute of Mathematical Sciences, 1996.

....finite element preconditioning, and additive Schwarz methods has been investigated by Pahl [28] Pavarino and Widlund [30] and Casarin [5] R nquist [33] and Casarin Present address: Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E mail: pff cfm.brown.edu [6] have studied iterative substructuring methods for spectral element solution of the fullycoupled steady Navier Stokes equations. R nquist also proposed a block Jacobi deflation based scheme applied to the consistent Poisson operator governing the pressure for the unsteady case [15, 32] The ....

M. Casarin, "Schwarz preconditioners for spectral and mortar finite element methods with applications to incompressible fluids", Ph.D. Thesis, Courant Institute of Math. Sci., NYU (1996).

....Ek T Ek M X i=1 T Omega i : 7) This operator does not exactly fit the Schwarz framework, but an analyisis similar to the proof of that result, together with a decomposition lemma involving the local spaces and bilinear forms just described, yield the following theorem. For the proof, see [Cas96]; cf. Bre94, Cai95] Theorem 4.1 The condition number of Tn satisfies: Tn ) C(1 log(N) 3 fi N : Remark 4.1 In three dimensions, edge and face functions play the role of the vertex and edge functions of the two dimensional version, respectively. For each edge, the edge function is the ....

.... and on the solution uN , and positive constants c(H 0 ) and C(H 0 ) such that the operator Q a = QH X s1 P s satisfies, 8u 2 P N 0;r ( Omega Gamma , and for H H 0 , a(Q a u; Q a u) C(H 0 )a(u; u) and c(H 0 )C Gamma2 0 a(u; u) a(Q a u; u) The proof of this result is given in [Cas96]. This estimate immediately implies an upper bound on the iteration count of the GMRES method applied to the preconditioned system Q a u N = b; where b is chosen so that u N is the vector of nodal values of uN . This result is an extension to the Navier Stokes equation of the Schwarz method for ....

Casarin M. A. (1996) Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids. PhD thesis, Courant Institute of Mathematical Sciences. Tech. Rep. 717, Department of Computer Science, Courant Institute.

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M. A. Casarin. Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids. PhD thesis, Courant Institute of Mathematical Sciences, March 1996. Tech. Rep. 717, Department of Computer Science, Courant Institute.

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M. A. Casarin Jr. Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids. PhD thesis, Courant Institute, New York University, February 1996, also Technical Report, Nr. 717, Department of Computer Science, Courant Institute.

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M. A. Casarin Jr. Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids. PhD thesis, Courant Institute, New York University, February 1996. Also Technical Report 717, Department of Computer Science, Courant Institute.

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