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1.
Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations.
When the subdomains are overlapping or nonoverlapping, these methods employ the optimal value of parameter(s) in the boundary
condition along the artificial interface to accelerate its convergence. In the literature, the analysis of optimized Schwarz
methods rely mostly on Fourier analysis and so the domains are restricted to be regular (rectangular). As in earlier papers,
the interface operator can be expressed in terms of Poincaré–Steklov operators. This enables the derivation of an upper bound
for the spectral radius of the interface operator on essentially arbitrary geometry. The problem of interest here is a PDE
with a discontinuous coefficient across the artificial interface. We derive convergence estimates when the mesh size h along the interface is small and the jump in the coefficient may be large. We consider two different types of Robin transmission
conditions in the Schwarz iteration: the first one leads to the best estimate when h is small, whereas for the second type, we derive a convergence estimate inversely proportional to the jump in the coefficient.
This latter result improves upon the rate of popular domain decomposition methods such as the Neumann–Neumann method or FETI-DP
methods, which was shown to be independent of the jump.
In memory of Gene Golub. 相似文献
2.
Mireille El Haddad Jos C. Garay Frdric Magouls Daniel B. Szyld 《Numerical Linear Algebra with Applications》2020,27(2)
Convergence of both synchronous and asynchronous optimized Schwarz algorithms for the shifted Laplacian operator on a bounded rectangular domain, in a one‐way subdivision of the computational domain, with overlap, is shown. Convergence results are obtained under very mild conditions on the size of the subdomains and on the amount of overlap. A couple of results are also given, relating the convergence rate of the asynchronous method to changes in the size of the domain. Numerical experiments illustrate the theoretical results. 相似文献
3.
Multilevel Schwarz methods for elliptic problems
with discontinuous coefficients in three dimensions
Summary.
Multilevel Schwarz methods are developed for a
conforming finite element approximation of second order elliptic problems. We
focus on problems in three dimensions with
possibly large jumps in the coefficients across the
interface separating the subregions. We establish
a condition number estimate for the iterative operator, which is
independent of the coefficients, and grows at most as the square
of the number of levels. We also characterize a class of distributions
of the coefficients,
called quasi-monotone, for which the weighted
-projection is
stable and for which we can use the standard piecewise
linear functions as a coarse space. In this case,
we obtain optimal methods, i.e. bounds which are independent of the number
of levels and subregions. We also design and analyze multilevel
methods with new coarse spaces
given by simple explicit formulas. We consider nonuniform meshes
and conclude by an analysis of multilevel iterative substructuring methods.
Received April 6, 1994 / Revised version received December 7,
1994 相似文献
4.
Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes 总被引:1,自引:0,他引:1
We give several additive Schwarz domain decomposition methods for solving finite element problems which arise from the discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Our theory requires no assumption (for the main results) on the substructures which constitute the whole domain, so each substructure can be of arbitrary shape and of different size. The global coarse mesh is allowed to be non-nested to the fine grid on which the discrete problem is to be solved and both the coarse meshes and the fine meshes need not be quasi-uniform. In this general setting, our algorithms have the same optimal convergence rate of the usual domain decomposition methods on structured meshes. The condition numbers of the preconditioned systems depend only on the (possibly small) overlap of the substructures and the size of the coares grid, but is independent of the sizes of the subdomains.Revised version on Sept. 20, 1994. Original version: CAM Report 93-40, Dec. 1993, Dept. of Math., UCLA.The work of this author was partially supported by the National Science Foundation under contract ASC 92-01266, the Army Research Office under contract DAAL03-91-G-0150, and ONR under contract ONR-N00014-92-J-1890.The work of this author was partially supported by the National Science Foundation under contract ASC 92-01266, the Army Research Office under contract DAAL03-91-G-0150, and subcontract DAAL03-91-C-0047. 相似文献
5.
Schwarz domain decomposition methods are developed for the numerical solution of singularly perturbed elliptic problems. Three
variants of a two-level Schwarz method with interface subproblems are investigated both theoretically and from the point of
view of their computer realization on a distributed memory multiprocessor computer. Numerical experiments illustrate their
parallel performance as well as their behavior with respect to the critical parameters such as the perturbation parameter,
the size of the interface subdomains and the number of parallel processors. Application of one of the methods to a model problem
with an interior layer of complex geometry is also discussed.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
6.
In this paper, we propose two variants of the additive Schwarz method for the approximation of second order elliptic boundary
value problems with discontinuous coefficients, on nonmatching grids using the lowest order Crouzeix-Raviart element for the
discretization in each subdomain. The overall discretization is based on the mortar technique for coupling nonmatching grids.
The convergence behavior of the proposed methods is similar to that of their closely related methods for conforming elements.
The condition number bound for the preconditioned systems is independent of the jumps of the coefficient, and depend linearly
on the ratio between the subdomain size and the mesh size. The performance of the methods is illustrated by some numerical
results.
This work has been supported by the Alexander von Humboldt Foundation and the special funds for major state basic research
projects (973) under 2005CB321701 and the National Science Foundation (NSF) of China (No.10471144)
This work has been supported in part by the Bergen Center for Computational Science, University of Bergen 相似文献
7.
Dominik Lellek 《Numerical Methods for Partial Differential Equations》2013,29(1):297-319
In this article, we are concerned with the numerical treatment of nonlinear elliptic boundary value problems. Our method of choice is a domain decomposition strategy. Partially following the lines from (Cohen, Dahmen and deVore, SIAM J Numer Anal 41 (2003), 1785–1823; Kappei, Appl Anal J Sci 90 (2011), 1323–1353; Lui, SIAM J Sci Comput 21 (2000), 1506–1523; Stevenson and Werner, Math Comp 78 (2009), 619–644), we develop an adaptive additive Schwarz method using wavelet frames. We show that the method converges with an asymptotically optimal rate and support our theoretical results with numerical tests in one and two space dimensions. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013 相似文献
8.
Marcus Sarkis 《Numerische Mathematik》1997,77(3):383-406
Summary. Two-level domain decomposition methods are developed for a simple nonconforming approximation of second order elliptic problems.
A bound is established for the condition number of these iterative methods, that grows only logarithmically with the number
of degrees of freedom in each subregion. This bound holds for two and three dimensions and is independent of jumps in the
value of the coefficients and number of subregions. We introduce face coarse spaces, and isomorphisms to map between conforming
and nonconforming spaces.
ReceivedMarch 1, 1995 / Revised version received January 16, 1996 相似文献
9.
Domain decomposition methods can be solved in various ways. In this paper, domain decomposition in strips is used. It is demonstrated that a special version of the Schwarz alternating iteration method coupled with coarse–fine‐mesh stabilization leads to a very efficient solver, which is easy to implement and has a behavior nearly independent of mesh and problem parameters. The novelty of the method is the use of alternating iterations between odd‐ and even‐numbered subdomains and the replacement of the commonly used coarse‐mesh stabilization method with coarse–fine‐mesh stabilization. 相似文献
10.
Three domain decomposition methods for saddle point problems are introduced and compared. The first two are block‐diagonal and block‐triangular preconditioners with diagonal blocks approximated by an overlapping Schwarz technique with positive definite local and coarse problems. The third is an overlapping Schwarz preconditioner based on indefinite local and coarse problems. Numerical experiments show that while all three methods are numerically scalable, the last method is almost always the most efficient. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
11.
A discontinuous Galerkin (DG) discretization of Dirichlet problem for second-order elliptic equations with discontinuous coefficients in 2-D is considered. For this discretization, balancing domain decomposition with constraints (BDDC) algorithms are designed and analyzed as an additive Schwarz method (ASM). The coarse and local problems are defined using special partitions of unity and edge constraints. Under certain assumptions on the coefficients and the mesh sizes across ∂Ωi, where the Ωi are disjoint subregions of the original region Ω, a condition number estimate C(1+maxilog(Hi/hi))2 is established with C independent of hi, Hi and the jumps of the coefficients. The algorithms are well suited for parallel computations and can be straightforwardly extended to the 3-D problems. Results of numerical tests are included which confirm the theoretical results and the necessity of the imposed assumptions. 相似文献
12.
On a nonoverlapping additive Schwarz method for h‐p discontinuous Galerkin discretization of elliptic problems
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Piotr Krzyżanowski 《Numerical Methods for Partial Differential Equations》2016,32(6):1572-1590
The condition number of a discontinuous Galerkin finite element discretization preconditioned with a nonoverlapping additive Schwarz method is analyzed. We improve the result of Antonietti and Houston (J Sci Comput 46 (2011), 124–149), where a bound has been proved for a two‐level nonoverlapping additive Schwarz method with coarse problem using polynomials of degree on a coarse mesh size . In a more general framework, where the concurrency of the algorithm is increased by applying solvers on subdomains smaller than the coarse grid cells, we prove that the condition number of the preconditioned system is where is the coarse space element degree polynomial and is the size of subdomain where local problems are solved in parallel. Our result also extends to the case of discontinuous coefficient, piecewise constant on the coarse grid, for a composite continuous–discontinuous Galerkin discretization. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1572–1590, 2016 相似文献
13.
Extrapolation methods can be a very effective technique used for accelerating the convergence of vector sequences. In this paper, these methods are used to accelerate the convergence of Schwarz iterative methods for nonlinear problems. A new implementation of the reduced-rank-extrapolation (RRE) method is introduced. Some convergence analysis is presented, and it is shown numerically that certain extrapolation methods can indeed be very effective in accelerating the convergence of Schwarz methods. 相似文献
14.
15.
Hongyi Yu 《Applied Numerical Mathematics》1999,30(4):475-491
We describe a method for solving parabolic partial differential equations (PDEs) using local refinement in time. Different time steps are used in different spatial regions based on a domain decomposition finite element method. Extrapolation methods based on either a linearly implicit mid-point rule or a linearly implicit Euler method are used to integrate in time. Extrapolation methods are a better fit than BDF methods in our context since local time stepping in different spatial regions precludes history information. Some linear and nonlinear examples demonstrate the effectiveness of the method. 相似文献
16.
S.H. Lui 《Numerical Algorithms》2002,30(1):59-69
The Schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. The full potential equation is derived from the Navier–Stokes equations assuming the fluid is compressible, inviscid, irrotational and isentropic. It is being used by the aircraft industry to model flow over an airfoil or even an entire aircraft. This paper shows that the additive and multiplicative versions of the Schwarz alternating method, when applied to the full potential equation in three dimensions, converge to the true solution geometrically. The assumptions are that the initial guess and the true solution are everywhere subsonic. We use the convergence proof by Tai and Xu and modify it for certain closed convex subsets. 相似文献
17.
Stephan Dahlke Dominik Lellek Rob Stevenson 《Numerical Functional Analysis & Optimization》2016,37(10):1213-1234
In this article, we are concerned with domain decomposition methods for the stationary incompressible Navier-Stokes equation. We construct an adaptive additive Schwarz method based on discretization by means of a divergence-free wavelet frame. We prove that the method is convergent and asymptotically optimal with respect to the degrees of freedom involved. 相似文献
18.
Ismael Herrera Robert Yates 《Numerical Methods for Partial Differential Equations》2001,17(5):495-517
Recently, Herrera presented a general theory of domain decomposition methods (DDM). This article is part of a line of research devoted to its further development and applications. According to it, DDM are classified into direct and indirect, which in turn can be subdivided into overlapping and nonoverlapping. Some articles dealing with general aspects of the theory and with indirect (Trefftz–Herrera) methods have been published. In the present article, a very general direct‐overlapping method, which subsumes Schwarz methods, is introduced. Also, this direct‐overlapping method is quite suitable for parallel implementation. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 495–517, 2001 相似文献
19.
Si studia, in un cilindro, il problema di Dirichlet per l'equazione ellittica del II ordine: , dove è l'operatore a coefficienti discontinui sull'asse x3 già introdotto da N. Ural'tseva per mostrare che l'equazione considerata può non avere soluzione nello spazio di Sobolev W2,p(p > 2) per qualche f?Lp. In questo lavoro si danno limitazioni a priori e teoremi di esistenza e unicità in W2,p quando p varia in un intervallo (p1(α), p2(α)), dipendente dalla costante di ellitticità α. Se p = p2(α) le limitazioni a priori cadono: l'esempio è quello di Ural'tseva. 相似文献
20.
Jan Van lent Robert Scheichl Ivan G. Graham 《Numerical Linear Algebra with Applications》2009,16(10):775-799
Two‐level overlapping Schwarz methods for elliptic partial differential equations combine local solves on overlapping domains with a global solve of a coarse approximation of the original problem. To obtain robust methods for equations with highly varying coefficients, it is important to carefully choose the coarse approximation. Recent theoretical results by the authors have shown that bases for such robust coarse spaces should be constructed such that the energy of the basis functions is minimized. We give a simple derivation of a method that finds such a minimum energy basis using one local solve per coarse space basis function and one global solve to enforce a partition of unity constraint. Although this global solve may seem prohibitively expensive, we demonstrate that a one‐level overlapping Schwarz method is an effective and scalable preconditioner and we show that such a preconditioner can be implemented efficiently using the Sherman–Morrison–Woodbury formula. The result is an elegant, scalable, algebraic method for constructing a robust coarse space given only the supports of the coarse space basis functions. Numerical experiments on a simple two‐dimensional model problem with a variety of binary and multiscale coefficients confirm this. Numerical experiments also show that, when used in a two‐level preconditioner, the energy‐minimizing coarse space gives better results than other coarse space constructions, such as the multiscale finite element approach. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献