J. Xu and J. Zou. Non--overlapping domain decomposition methods. Technical report, Mathematics Department, Pennsylvania State University, University Park, PA, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....methods [4] There are two principal viewpoints of non overlapping methods, preconditioning and interface relaxation. For an in depth and up to date survey of non overlapping domain decomposition methods considered and analyzed from the preconditioning viewpoint the reader is referred to [13] and for a general formulation and analysis of interface relaxation methods to [8] We give a brief presentation of the interface relaxation method philosophy and practice, in order to identify its main characteristics. Interface relaxation is a step beyond non overlapping domain decomposition; ....

J. Xu and J. Zou. Non--overlapping domain decomposition methods. Technical report, Mathematics Department, Pennsylvania State University, University Park, PA, 1996.

....need to be solved on the decomposed subspaces. We also emphasis that our approach is valid for general space decomposition techniques. So the applications is not restricted to domain decomposition and multigrid methods. Other space decomposition techniques can also be considered, see [SBG96] [XZ98]. The two algorithms given in this work were first proposed in [Tai92] see also [Tai94] Tai95a] Tai95c] and [TE98] where the qualitative convergence of the algorithms was proved, but the uniform rate of convergence was not given there. 1.2 Optimization problems and subspace correction ....

Xu J. and Zou J. (1998) Nonoverlapping domain decomposition methods. SIAM Rev. pages 857--914.

....for later parallelization. Based on this principle a series of algorithms has been developed, i.e. overlapping Schwarz methods, non overlapping Schur complement methods, etc. The development of these methods up to now can be found in the conference proceedings [59] see also the surveys [51, 148, 169]. 2.4.1 Overlapping Schwarz methods The first domain decomposition methods for the solution of PDEs were the overlapping Schwarz methods, named after H. A. Schwarz [145] They were used for the existence proof of a continuous solution of the Poisson equation in a domain, which was composed of the ....

J. Xu and J. Zou, Non-overlapping domain decomposition methods, SIAM Rev., (1996). submitted.

....problem involves a Schur complement matrix at the discrete level, and a Steklov Poincare operator at the continuous level. Substructuring is an efficient, parallel direct method, but requires good and possibly expensive and complicated preconditioners for the iterative interface problem solver [39]. Nonoverlapping Schwarz alternating methods are 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] ....

....the electrostatic potential has jumps in the form u 1 1 A ffl out ffl in u 2 2 A = j on Gamma; 2.5) where ffl out and ffl in are the uniform dielectric constant outside and inside the molecule, respectively. Schwarz overlapping and substructuring domain decomposition methods [5, 11, 12, 13, 17, 24, 26, 37, 38, 39] have been analyzed for the special linear case (2.1) 2.4) in which = 0 and j = 0: It is not clear that any of these methods applies directly to the general case with 6= 0 and j 6= 0: In [27] a finite difference method without domain decomposition was considered for the problem (2.1) 2.4) ....

J. Xu and J. Zou, Nonoverlapping domain decomposition methods, to appear.

....be obtained by the techniques used there. A second ingredient needed for the convergence proof is the existence of a bounded hp extension or lifting operator. Such operators have been given for the UNIFORM hp CONVERGENCE RESULTS FOR THE MORTAR FEM 7 p version (e.g. 3, 4] the h version (e.g. [25, 26]) and the hp version with quasiuniform meshes (e.g. 4] but not for the hp version with general non quasiuniform meshes. For this, we have the following theorem. Theorem 3.2. For each fl 2 Z and i such that fl ae Omega i and i = NM(fl) there exists an extension operator R i;fl = R h;k ....

....Then given z 2 S h;k ( Gamma) there exists v 2 V h;k( Omega 0 ) satisfying, for any ffl 0, v = z on Gamma; jjvjj 1 Cjjzjj 1 2 ffl; Gamma (5.1) with C a constant independent of z; h; k but depending on ffl. We start with a technical lemma whose proof is adapted from Section 3:2:2 of [26]. Lemma 5.1. Let z 2 S h;k ( Gamma) Then there exists v 2 V h;k( Omega 0 ) satisfying, v = z on Gamma; jjvjj 1; Omega 0 Cjjzjj 1 2 ; Gamma (5.2) where C is a constant independent of h but depending on k. Proof. Let V be the extension of z satisfying, Gamma DeltaV V = 0 in Omega 0 ; ....

J. Xu and J. Zou. Non-overlapping domain decomposition methods. Submitted to SIAM Review.

....solved on the decomposed subspaces. We also emphasis that our approach is valid for general space decomposition techniques. So the applications is not restricted to domain decomposition and multigrid methods. Other space decomposition techniques can also be considered, see [10] 23, p. 169 184] [31]. The two algorithms given in this work were rst proposed in [24] see also [25] 27] and [28] where the convergence of the algorithms was proved, but the rate of convergence was not given. Uniform convergence was proved for a special case in [29] 2 Optimization problems and subspace ....

J. Xu and J. Zou. Nonoverlapping domain decomposition methods. Technical report, Pennsylvania State University, Department of Mathetics, 1996.

....solved on the decomposed subspaces. We also emphasis that our approach is valid for general space decomposition techniques. So the applications is not restricted to domain decomposition and multigrid methods. Other space decomposition techniques can also be considered, see [10] 23, p. 169 184] [34]. The two algorithms given in this work were first proposed in [24] see also [25] 26] 27] 28] and [29] where the qualitative convergence of the algorithms was proved, but the uniform rate of convergence was not given there. 2 Optimization problems and subspace correction methods Consider ....

J. Xu and J. Zou. Non-overlapping domain decomposition methods. SIAM Review (to appear).

....1) this with (3.20) and the triangle inequality implies the required result for Pi 0 h . Cl ement s interpolant. We now introduce a locally defined interpolant R 0 H proposed by Cl ement in [15] Operators with similar properties to R 0 H can be found in Scott Zhang [27] see also Xu Zou [34]. Definition 3.1. The mapping R 0 H : L 2( Omega 0 ) V 0 and RH : L 2( Omega 0 ) V 0 are defined by R 0 H u = X q H i 2 Omega 0 Q i u(q H i ) H i ; RH u = X q H i 2 Omega 0 Q i u(q H i ) H i ; 8 u 2 L 2( Omega 0 ) where Q i u 2 P 1 (O H i ) O H i = supp ....

J. Xu and J. Zou. Non-overlapping domain decomposition methods. Submitted to SIAM Review.

....shape but is smooth. The resultant linear systems are always symmetric and positive definite when the original PDEs are self adjoint and uniformly elliptic. And in particular, the domain decomposition methods, which have been investigated widely in recent years (cf. Chan Zou [5] and Xu Zou [26]) can be applied here to construct efficient preconditioned iterative methods for solving these large scale and sparse linear systems of equations. And different from the previous finite element methods, the calculations of the stiffness matrix and the interface integral related to the jumps of ....

J. Xu and J. Zou. Non-overlapping domain decomposition methods. submitted.

....for Pi 0 h . T. Chan and J. Zou Convergence of Unstructured Multilevel Methods 15 Cl ement s interpolant We now introduce a locally defined interpolant R 0 H proposed by Cl ement in [18] Operators with similar properties to R 0 H can be found in Scott Zhang [32] see also Xu Zou [39]. Definition 5 The mapping R 0 H : L 2( Omega 0 ) V 0 and RH : L 2( Omega 0 ) V 0 are defined by R 0 H u = X q H i 2 Omega 0 Q i u(q H i ) H i ; RH u = X q H i 2 Omega 0 Q i u(q H i ) H i ; 8u 2 L 2( Omega 0 ) where Q i u 2 P 1 (O H i ) O H i = supp ....

J. Xu and J. Zou, Non-overlapping domain decomposition methods, MATH-96-19 (93), Department of Mathematics, The Chinese University of Hong Kong, 1996.

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