Cheng H. (1993) Iterative Solution of Elliptic Finite Element Problems on Partially Refined Meshes and the Effect of Using Inexact Solvers. PhD thesis, Courant Institute of Mathematical Science, New York University.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the spectral equivalence inequalities fl C CC SC fl C CC and fl I C I K I fl I C I : 1.5) If we have a constant c E so that fl fl fl fl v C B IC v C fl fl fl fl K c E k v C k S C 8v C 2 R Nc (1. 6) holds then the upper and lower bounds of the condition number (C Gamma1 K) [HLM91, Che93] can be estimated as O(c 2 E ) C Gamma1 K) O(c 4 E ) 1.7) MULTILEVEL EXTENSION TECHNIQUES IN DD 361 In the remaining chapter we construct extension techniques defining B IC;i which are cheap to implement and result in a constant c E independent of or slightly dependent on the ....

....proved that the new extension method is also successfully applicable to practical problems. The implementation and theoretical analysis of the BPX like extension together with smoothing sweeps will be done in a forthcoming paper. Also, a comparison to other extension techniques proposed in [Nep91, Che93, BPV96] should be done on a more challenging example, with respect to the CPU time needed to solve (1.1) ....

Cheng H. (1993) Iterative Solution of Elliptic Finite Element Problems on Partially Refined Meshes and the Effect of Using Inexact Solvers. PhD thesis, Courant Institute of Mathematical Science, New York University.

....improves the extension constant c E . All these components have optimal arithmetic complexity, i.e. the operations E, E T and C Gamma1 I need O(N) operations, while the application of C Gamma1 C needs O(N C ) operations only. The precise analysis of the overall operator C is given in [13] [6]. In [12] it is shown, how a symmetric multiplicative Schwarz preconditioner fits into the framework of additive Schwarz preconditioners. 4 The Projection Method with DD Preconditioning In this section we will apply the DD preconditioner for the projection method. The first two requirements, ....

H. Cheng. Iterative Solution of Elliptic Finite Element Problems on Partially Refined Meshes and the Effect of Using Inexact Solvers. PhD thesis, New York University, Courant Institute of Mathematical Sciences, New York, 1993.

.... spectral equivalence inequalities fl C CC SC fl C CC and fl I C I K I fl I C I : 5) If we have a constant c E so that fl fl fl fl v C B IC v C fl fl fl fl K c E k v C k S C 8v C 2 R Nc (6) holds then the upper and lower bounds of the condition number (C Gamma1 K) HLM91, Che93] can be estimated as O(c 2 E ) C Gamma1 K) O(c 4 E ) 7) In the remaining paper we construct extension techniques defining B IC;i which are cheap to implement and should result in a constant c E independent of or slightly dependent on the discretization parameter h, respectively the ....

....in a faster solver when the algorithmic improvement from 4.3 is used, see also [Haa96] The implementation and theoretical analysis of the BPX like extension together with smoothing sweeps will be done in a forthcoming paper. Also, a comparison to other extension techniques proposed in [Nep91, Che93, BPV96] should be done on a more challenging example with respect to the CPU time needed to solve (1) 9 ....

Cheng H. (1993) Iterative Solution of Elliptic Finite Element Problems on Partially Refined Meshes and the Effect of Using Inexact Solvers. PhD thesis, Courant Institute of Mathematical Science, New York University.

.... equivalence inequalities fl C C C S C fl C C C and fl I C I K I fl I C I : If we have additionally a constant c E so that fl fl fl fl v C B IC v C fl fl fl fl K c E k v C k S C 8v C 2 R Nc (23) holds then the upper and lower bounds of the condition number (C Gamma1 K) [13, 8] can be estimated as O(c 2 E ) C Gamma1 K) O(c 4 E ) 24) Estimate (23) represents the result from Theorem 1 in a discrete sense, so that B IC can be chosen as the discrete extension operator defined in (14) 19) or (20) Additionally, Algorithm 1 requires B CI = B T IC so that the ....

H. Cheng. Iterative Solution of Elliptic Finite Element Problems on Partially Refined Meshes and the Effect of Using Inexact Solvers. PhD thesis, Courant Institute of Mathematical Science, New York University, 1993.

....note, in particular, that solutions of elliptic problems with highly discontinuous coefficients are very likely to become increasingly singular when we approach the wire basket. Nested local refinements have previously been analyzed by Bornemann and Yserentant [3] Bramble and Pasciak [5] Cheng [25, 26], Oswald [66, 67] and Yserentant [92, 93] By nested local refinement we mean that an element, which is not refined at level j, cannot be a candidate for further refinement. Under certain assumptions on the local refinement, optimal multilevel preconditioners previously have been obtained for ....

Hsuanjen Cheng. Iterative Solution of Elliptic Finite Element Problems on Partially Refined Meshes and the Effect of Using Inexact Solvers. PhD thesis, Courant Institute of Mathematical Sciences, October 1993. Tech. Rep. 649, Department of Computer Science, Courant Institute.

....note, in particular, that solutions of elliptic problems with highly discontinuous coefficients are very likely to become increasingly singular when we approach the wire basket. Nested local refinements have previously been analyzed by Bornemann and Yserentant [1] Bramble and Pasciak [2] Cheng [6,7], Oswald [18,19] and Yserentant [27,28] By nested local refinement we mean that an element, which is not refined at level j, cannot be a candidate for further refinement. Under certain assumptions on the local refinement, optimal multilevel preconditioners have been obtained for problems with ....

Hsuanjen Cheng. Iterative Solution of Elliptic Finite Element Problems on Partially Refined Meshes and the Effect of Using Inexact Solvers. PhD thesis, Courant Institute of Mathematical Sciences, October 1993. Tech. Rep. 649, Department of Computer Science, Courant Institute.

....the spectral equivalence inequalities fl C CC SC fl C CC and fl I C I K I fl I C I : 1.5) If we have a constant c E so that fl fl fl fl v C B IC v C fl fl fl fl K c E k v C k S C 8v C 2 R Nc (1. 6) holds then the upper and lower bounds of the condition number (C Gamma1 K) [HLM91, Che93] can be estimated as O(c 2 E ) C Gamma1 K) O(c 4 E ) 1.7) EXTENSION TECHNIQUES IN DD 3 In the remaining chapter we construct extension techniques defining B IC;i which are cheap to implement and should result in a constant c E independent of or slightly dependent on the ....

....proved that the new extension method is also successfully applicable to practical problems. The implementation and theoretical analysis of the BPX like extension together with smoothing sweeps will be done in a forthcoming paper. Also, a comparison to other extension techniques proposed in [Nep91, Che93, BPV96] should be done on a more challenging example with respect to the CPU time needed to solve (1.1) ....

Cheng H. (1993) Iterative Solution of Elliptic Finite Element Problems on Partially Refined Meshes and the Effect of Using Inexact Solvers. PhD thesis, Courant Institute of Mathematical Science, New York University.

.... equivalence inequalities fl C C C S C fl C C C and fl I C I K I fl I C I : If we have additionally a constant c E so that fl fl fl fl v C B IC v C fl fl fl fl K c E k v C k S C 8v C 2 R Nc (23) holds then the upper and lower bounds of the condition number (C Gamma1 K) [12, 7] can be estimated as O(c 2 E ) C Gamma1 K) O(c 4 E ) 24) Estimate (23) represents the result from Theorem 1 in a discrete sense, so that B IC can be chosen as the discrete extension operator defined in (14, 19, 20) Additionally, Algorithm 1 requires B CI = B T IC so that the ....

H. Cheng. Iterative Solution of Elliptic Finite Element Problems on Partially Refined Meshes and the Effect of Using Inexact Solvers. PhD thesis, Courant Institute of Mathematical Science, New York University, 1993.

....of the composite model is computed. We use the framework of multilevel additive Schwarz methods, which is described in Dryja and Widlund [7] and Zhang [18] to develop a new kind of algorithm for composite finite element problems. If we compare this kind of algorithm with the AFAC methods in [4], we can see that they are both additive Schwarz methods. In the new algorithms, we decompose the problems corresponding to the refined subregions with uniform mesh size, used in AFAC, into many much smaller problems which are much easier to solve. However, this is at the expense of slower ....

....do not need to use the fact that the quasi interpolants are bounded linear mappings in the space H 1 0 ( Omega Gamma1 9 In order to prove the main lemmas in this section, we need the following four lemmas. The proof of the first lemma can be carried out using a standard duality argument; cf. [4]. The proof of the second lemma is based on using smooth functions to approximate elements in H 1( Omega Gamma and applying the fundamental theorem of calculus to the region. Proof of Lemma 3 may be found in [10] and [11] Proofs of Lemma 4 may be found in [9] and [16] Lemma 1. Let P l be ....

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H. Cheng, Iterative Solution of Elliptic Finite Element Problems on Partially Refined Meshes and the Effect of Using Inexact Solvers, PhD thesis, Courant Institute of Mathematical Sciences, October 1993. Tech. Rep. 649, Department of Computer Science, Courant Institute.

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