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1.
Summary.
In recent years, it has been shown that many modern iterative algorithms
(multigrid schemes, multilevel preconditioners, domain decomposition
methods etc.)
for solving problems resulting from the discretization
of PDEs can be
interpreted as additive (Jacobi-like) or multiplicative
(Gauss-Seidel-like) subspace correction methods. The key to their
analysis is the study of certain metric properties of the underlying
splitting of the discretization space into a sum of subspaces
and the splitting of the variational problem on into auxiliary problems on
these subspaces.
In this paper, we propose a modification of the abstract convergence
theory of the additive and multiplicative Schwarz methods, that
makes the relation to traditional iteration methods more explicit.
The analysis of the additive and multiplicative Schwarz iterations
can be carried out in almost the same spirit as in the
traditional block-matrix
situation, making convergence proofs of multilevel and domain decomposition
methods clearer, or, at least, more classical.
In addition, we present a
new bound for the convergence rate of the appropriately scaled
multiplicative Schwarz method directly in terms
of the condition number of the corresponding additive
Schwarz operator.
These results may be viewed as an appendix to the
recent surveys [X], [Ys].
Received February 1, 1994 / Revised version received August
1, 1994 相似文献
2.
Summary. We study some additive Schwarz algorithms for the version Galerkin boundary element method applied to some weakly singular and hypersingular integral equations of the first
kind. Both non-overlapping and overlapping methods are considered. We prove that the condition numbers of the additive Schwarz
operators grow at most as independently of h, where p is the degree of the polynomials used in the Galerkin boundary element schemes and h is the mesh size. Thus we show that additive Schwarz methods, which were originally designed for finite element discretisation
of differential equations, are also efficient preconditioners for some boundary integral operators, which are non-local operators.
Received June 15, 1997 / Revised version received July 7, 1998 / Published online February 17, 2000 相似文献
3.
Xuejun Zhang 《Numerische Mathematik》1992,63(1):521-539
Summary We consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations. A class of multilevel algorithms are studied using the additive Schwarz framework. We establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels. This is an improvement on Dryja and Widlund's result on a multilevel additive Schwarz algorithm, as well as Bramble, Pasciak and Xu's result on the BPX algorithm. Some multiplicative variants of the multilevel methods are also considered. We establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one, which is independent of the number of levels. For a proper ordering, the iteration operators correspond to the error propagation operators of certain V-cycle multigrid methods, using Gauss-Seidel and damped Jacobi methods as smoothers, respectively.This work was supported in part by the National Science Foundation under Grants NSF-CCR-8903003 at Courant Institute of Mathematical Sciences, New York University and NSF-ASC-8958544 at Department of Computer Science, University of Maryland. 相似文献
4.
带非线性源项的变分不等式的区域分解法及其收敛速度分析 总被引:4,自引:0,他引:4
本文考虑一类带非线性源项的变化不等式。针对其有限元离散问题,我们构造了乘性与加性Schwarz算法,其产生的上解序列或下解序列不仅单调收敛于有限元解,而且具有限元网格h无关的收敛率. 相似文献
5.
Summary. We consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational
problems posed in the Hilbert spaces and in three dimensions. We show that if appropriate finite element spaces and appropriate additive or multiplicative Schwarz
smoothers are used, then the multigrid V-cycle is an efficient solver and preconditioner for the discrete operator. All results
are uniform with respect to the mesh size, the number of mesh levels, and weights on the two terms in the inner products.
Received June 12, 1998 / Revised version received March 12, 1999 / Published online January 27, 2000 相似文献
6.
For any continuous bilinear form defined on a pair of Hilbert spaces satisfying the compatibility Ladyshenskaya–Babušca–Brezzi condition, symmetric Schur complement operators can be defined on each of the two Hilbert spaces. In this paper, we find bounds for the spectrum of the Schur operators only in terms of the compatibility and continuity constants. In light of the new spectral results for the Schur complements, we review the classical Babušca–Brezzi theory, find sharp stability estimates, and improve a convergence result for the inexact Uzawa algorithm. We prove that for any symmetric saddle point problem, the inexact Uzawa algorithm converges, provided that the inexact process for inverting the residual at each step has the relative error smaller than 1/3. As a consequence, we provide a new type of algorithm for discretizing saddle point problems, which combines the inexact Uzawa iterations with standard a posteriori error analysis and does not require the discrete stability conditions. 相似文献
7.
Summary. Multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline
collocation with piecewise Hermite bicubics is applied to the Dirichlet boundary value problem for a self-adjoint elliptic
partial differential equation on a rectangle. Additive and multiplicative preconditioners are defined respectively as sums
and products of independent operators on a sequence of nested subspaces of the fine partition approximation space. A general
theory of additive and multiplicative Schwarz methods is used to prove that the preconditioners are spectrally equivalent
to the collocation discretization of the Laplacian with the spectral constants independent of the fine partition stepsize
and the number of levels. The preconditioned conjugate gradient and preconditioned Orthomin methods are considered for the
solution of collocation problems. An implementation of the methods is discussed and the results of numerical experiments are
presented.
Received March 1, 1994 / Revised version received January 23, 1996 相似文献
8.
An additive Schwarz method for variational inequalities 总被引:3,自引:0,他引:3
This paper proposes an additive Schwarz method for variational inequalities and their approximations by finite element methods. The Schwarz domain decomposition method is proved to converge with a geometric rate depending on the decomposition of the domain. The result is based on an abstract framework of convergence analysis established for general variational inequalities in Hilbert spaces.
9.
Luca F. Pavarino 《Numerische Mathematik》1993,66(1):493-515
Summary In this paper, we study some additive Schwarz methods (ASM) for thep-version finite element method. We consider linear, scalar, self adjoint, second order elliptic problems and quadrilateral elements in the finite element discretization. We prove a constant bound independent of the degreep and the number of subdomainsN, for the condition number of the ASM iteration operator. This optimal result is obtained first in dimension two. It is then generalized to dimensionn and to a variant of the method on the interface. Numerical experiments confirming these results are reported. As is the case for other additive Schwarz methods, our algorithms are highly parallel and scalable.This work was supported in part by the Applied Math. Sci. Program of the U.S. Department of Energy under contract DE-FG02-88ER25053 and, in part, by the National Science Foundation under Grant NSF-CCR-9204255 相似文献
10.
Summary. We compare additive and multiplicative Schwarz preconditioners for the iterative solution of regularized linear inverse problems,
extending and complementing earlier results of Hackbusch, King, and Rieder. Our main findings are that the classical convergence
estimates are not useful in this context: rather, we observe that for regularized ill-posed problems with relevant parameter
values the additive Schwarz preconditioner significantly increases the condition number. On the other hand, the multiplicative version greatly improves conditioning, much beyond the existing
theoretical worst-case bounds.
We present a theoretical analysis to support these results, and include a brief numerical example. More numerical examples
with real applications will be given elsewhere.
Received May 28, 1998 / Published online: July 7, 1999 相似文献
11.
Summary For solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed. The methods are within the general framework of the product (multiplicative) scheme for operators in a Hilbert space, proposed recently by Bramble, Pasciak, Wang, and Xu [5,6,26,27] and make use of certain multilevel decomposition of the corresponding spaces for the flux variable. 相似文献
12.
In this paper, multiplicative and additive generalized Schwarz algorithms for solving obstacle problems with elliptic operators
are developed and analyzed. Compared with the classical Schwarz algorithms, in which the subproblems are coupled by the Dirichlet
boundary conditions, the generalized Schwarz algorithms use Robin conditions with parameters as the transmission conditions
on the interface boundaries. As a result, the convergence rate can be speeded up by choosing Robin parameters properly. Convergence
of the algorithms is established.
This work was supported by 973 national project of China (2004CB719402) and by national nature science foundation of China
(10671060). 相似文献
13.
We consider the multiplicative and additive Schwarz methods for solving linear systems of equations and we compare their asymptotic rate of convergence. Moreover, we compare the multiplicative Schwarz method with the weighted restricted additive Schwarz method. We prove that the multiplicative Schwarz method is the fastest method among these three. Our comparisons can be done in the case of exact and inexact subspaces solves. In addition, we analyse two ways of adding a coarse grid correction – multiplicatively or additively.
Mathematics Subject Classification (1991):65F10, 65F35, 65M55 相似文献
14.
Xiao-Chuan Cai 《Numerische Mathematik》1991,60(1):41-61
Summary In this paper, we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems. We introduce three additive Schwarz type domain decomposition methods for general, not necessarily selfadjoint, linear, second order, parabolic partial differential equations and also study the convergence rates of these algorithms. The resulting preconditioned linear system of equations is solved by the generalized minimal residual method. Numerical results are also reported.This work was supported in part by the National Science Foundation under Grant NSF-CCR-8903003 at the Courant Institute, New York University and in part by the National Science Foundation under contract number DCR-8521451 and ECS-8957475 at Yale University 相似文献
15.
I. P. Ryazantseva 《Differential Equations》2008,44(7):1006-1017
For quasivariational inequalities of special form in a Hilbert space, we construct a continuous second-order method and a discrete version of this method, prove the strong convergence of these methods, and indicate the possibility of obtaining estimates for the convergence rate. We separately study the convergence of the continuous method under the assumption that the operators describing the quasivariational inequality to be solved are potential. We establish sufficient conditions for the unique solvability of the nonlinear problems determining these methods. 相似文献
16.
Jens Flemming 《Numerical Functional Analysis & Optimization》2013,34(3):254-284
This article addresses Tikhonov-like regularization methods with convex penalty functionals for solving nonlinear ill-posed operator equations formulated in Banach or, more general, topological spaces. We present an approach for proving convergence rates that combines advantages of approximate source conditions and variational inequalities. Precisely, our technique provides both a wide range of convergence rates and the capability to handle general and not necessarily convex residual terms as well as nonsmooth operators. Initially formulated for topological spaces, the approach is extensively discussed for Banach and Hilbert space situations, showing that it generalizes some well-known convergence rates results. 相似文献
17.
Tarek P. Mathew 《Numerische Mathematik》1993,65(1):469-492
Summary In this paper we discuss bounds for the convergence rates of several domain decomposition algorithms to solve symmetric, indefinite linear systems arising from mixed finite element discretizations of elliptic problems. The algorithms include Schwarz methods and iterative refinement methods on locally refined grids. The implementation of Schwarz and iterative refinement algorithms have been discussed in part I. A discussion on the stability of mixed discretizations on locally refined grids is included and quantiative estimates for the convergence rates of some iterative refinement algorithms are also derived.Department of Mathematics, University of Wyoming, Laramie, WY 82071-3036. This work was supported in part by the National Science Foundation under Grant NSF-CCR-8903003, while the author was a graduate student at New York University, and in part by NSF Grant ASC 9003002, while the author was a Visiting, Assistant Researcher at UCLA. 相似文献
18.
Norbert Heuer 《Numerische Mathematik》1998,79(3):371-396
Summary. We study preconditioners for the -version of the boundary element method for hypersingular integral equations in three dimensions. The preconditioners are
based on iterative substructuring of the underlying ansatz spaces which are constructed by using discretely harmonic basis
functions. We consider a so-called wire basket preconditioner and a non-overlapping additive Schwarz method based on the complete
natural splitting, i.e. with respect to the nodal, edge and interior functions, as well as an almost diagonal preconditioner.
In any case we add the space of piecewise bilinear functions which eliminate the dependence of the condition numbers on the
mesh size. For all these methods we prove that the resulting condition numbers are bounded by . Here, is the polynomial degree of the ansatz functions and is a constant which is independent of and the mesh size of the underlying boundary element mesh. Numerical experiments supporting these results are reported.
Received July 8, 1996 / Revised version received January 8, 1997 相似文献
19.
The rates of convergence of two Schwarz alternating methods are analyzed for the iterative solution of a discrete problem which arises when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet problem for Poisson's equation on a rectangle. In the first method, the rectangle is divided into two overlapping subrectangles, while three overlapping subrectangles are used in the second method. Fourier analysis is used to obtain explicit formulas for the convergence factors by which theH
1-norm of the errors is reduced in one iteration of the Schwarz methods. It is shown numerically that while these factors depend on the size of overlap, they are independent of the partition stepsize. Results of numerical experiments are presented which confirm the established rates of convergence of the Schwarz methods.This research was supported in part by funds from the National Science Foundation grant CCR-9103451. 相似文献
20.
In this paper, we introduce two new iterative algorithms for finding a common element of the set of fixed points of a quasi-nonexpansive mapping and the set of solutions of the variational inequality problem with a monotone and Lipschitz continuous mapping in real Hilbert spaces, by combining a modified Tseng’s extragradient scheme with the Mann approximation method. We prove weak and strong convergence theorems for the sequences generated by these iterative algorithms. The main advantages of our algorithms are that the construction of solution approximations and the proof of convergence of the algorithms are performed without the prior knowledge of the Lipschitz constant of cost operators. Finally, we provide numerical experiments to show the efficiency and advantage of the proposed algorithms. 相似文献