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126
Schwarz Analysis Of Iterative Substructuring Algorithms For Elliptic Problems In Three Dimensions
 SIAM J. Numer. Anal
, 1993
"... . Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations of partial differential equations. The preconditioners are constructed from exact or approximate sol ..."
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Cited by 110 (26 self)
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. Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations of partial differential equations. The preconditioners are constructed from exact or approximate solvers for the same partial differential equation restricted to a set of subregions into which the given region has been divided. In addition, the preconditioner is often augmented by a coarse, second level approximation, that provides additional, global exchange of information, and which can enhance the rate of convergence considerably. The iterative substructuring methods, based on decompositions of the region into nonoverlapping subregions, form one of the main families of such algorithms. Many domain decomposition algorithms can conveniently be described and analyzed as Schwarz methods. These algorithms are fully defined in terms of a set of subspaces and auxiliary bilinear forms. A gener...
A comparison of some domain decomposition and ILU preconditioned iterative methods for nonsymmetric elliptic problems
 Numer. Linear Algebra Appl
, 1994
"... In recent years, competitive domaindecomposed preconditioned iterative techniques have beendeveloped for nonsymmetric elliptic problems. In these techniques, a large problem is divided into many smaller problems whose requirements for coordination can be controlled to allow e ective solution on par ..."
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Cited by 55 (13 self)
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In recent years, competitive domaindecomposed preconditioned iterative techniques have beendeveloped for nonsymmetric elliptic problems. In these techniques, a large problem is divided into many smaller problems whose requirements for coordination can be controlled to allow e ective solution on parallel machines. Acentral question is how tochoose these small problems and how to arrange the order of their solution. Di erent speci cations of decomposition and solution order lead to a plethora of algorithms possessing complementary advantages and disadvantages. In this report we compare several methods, including the additive Schwarz algorithm, the classical multiplicative Schwarz algorithm, an accelerated multiplicative Schwarz algorithm, the tile algorithm, the CGK algorithm, the CSPD algorithm, and also the popular global ILUfamily of preconditioners, on some nonsymmetric or inde nite elliptic model problems discretized by nite di erence methods. The preconditioned problems are solved by the unrestarted GMRES method. A version of the accelerated multiplicative Schwarz method is a consistently good performer.
Overlapping Schwarz Methods On Unstructured Meshes Using NonMatching Coarse Grids
 Numer. Math
, 1996
"... . We consider two level overlapping Schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Standard finite element interpolation from the coarse to the fine grid may ..."
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Cited by 49 (17 self)
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. We consider two level overlapping Schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Standard finite element interpolation from the coarse to the fine grid may be used. Our theory requires no assumption 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 nonnested 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 quasiuniform. In addition, the domains defined by the fine and coarse grid need not be identical. The one important constraint is that the closure of the coarse grid must cover any portion of the fine grid boundary for which Neumann boundary conditions are given. In this general setting, our algorithms have the same optimal convergence rate of the usual ...
Domain decomposition algorithms for indefinite elliptic problems
 SIAM J. Sci. Stat. Comput
, 1992
"... Iterative methods for linear systems of algebraic equations arising from the finite element discretization of nonsymmetric and indefinite elliptic problems are considered. Methods previously known to work well for positive definite, symmetric problems are extended to certain nonsymmetric problems, w ..."
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Cited by 48 (16 self)
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Iterative methods for linear systems of algebraic equations arising from the finite element discretization of nonsymmetric and indefinite elliptic problems are considered. Methods previously known to work well for positive definite, symmetric problems are extended to certain nonsymmetric problems, which can also have some eigenvalues in the left half plane. We first consider an additive Schwarz method applied to linear, second order, symmetric or nonsymmetric, indefinite elliptic boundary value problems in two and three dimensions. An alternative linear system, which has the same solution as the original problem, is derived and this system is then solved by using GMRES, an iterative method of conjugate gradient type. In each iteration step, a coarse mesh finite element problem and a number of local problems are solved on small, overlapping subregions into which the original region is subdivided. We show that the rate of convergence is independent of the number of degrees of freedom and the number of local problems if the coarse mesh is fine enough. The performance of the method in two dimensions is illustrated by results of several numerical experiments. We also consider two other iterative method for solving the same class of elliptic problems in two dimensions. Using an observation of Dryja and Widlund, we show that the rate of convergence of certain iterative substructuring methods deteriorates only quite slowly when the local problems increase in size. A similar result is established for Yserentant’s hierarchical basis method.
Optimized Schwarz methods
 SIAM Journal on Numerical Analysis
, 2006
"... Abstract. Optimized Schwarz methods are a new class of Schwarz methods with greatly enhanced convergence properties. They converge uniformly faster than classical Schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical Schwarz methods if th ..."
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Cited by 47 (11 self)
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Abstract. Optimized Schwarz methods are a new class of Schwarz methods with greatly enhanced convergence properties. They converge uniformly faster than classical Schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical Schwarz methods if the overlap is of the order of the mesh parameter, which is often the case in practical applications. They achieve this performance by using new transmission conditions between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem. We analyze in this paper these new methods for symmetric positive definite problems and show their relation to other modern domain decomposition methods like the new Finite Element Tearing and Interconnect (FETI) variants.
Convergence of a Balancing Domain Decomposition by Constraints and Energy Minimization
, 2002
"... A convergence theory is presented for a substructuring preconditioner based on constrained energy minimization concepts. The preconditioner is formulated as an Additive Schwarz method and analyzed by building on existing results for Balancing Domain Decomposition. The main result is a bound on the c ..."
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Cited by 46 (9 self)
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A convergence theory is presented for a substructuring preconditioner based on constrained energy minimization concepts. The preconditioner is formulated as an Additive Schwarz method and analyzed by building on existing results for Balancing Domain Decomposition. The main result is a bound on the condition number based on inequalities involving the matrices of the preconditioner. Estimates of the usual form C(1 + log²(H/h)) are obtained under the standard assumptions of substructuring theory. Computational results demonstrating the performance of method are included.
Additive Schwarz algorithms for parabolic convectiondiffusion equations
 Numer. Math
, 1991
"... 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 equ ..."
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Cited by 41 (6 self)
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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. Key words Schwarz’s alternating method, domain decomposition, parabolic convectiondiffusion equation, finite elements. AMS(MOS) subject classifications. 65N30, 65F10 1
Optimized Schwarz methods without overlap for the Helmholtz equation
 SIAM J. Sci. Comput
"... Abstract. The classical Schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel. Convergence is achieved through overlap of the subdomains. We study in this paper a variant of the Schwarz method which converges without overlap for the Helmholtz eq ..."
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Cited by 37 (12 self)
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Abstract. The classical Schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel. Convergence is achieved through overlap of the subdomains. We study in this paper a variant of the Schwarz method which converges without overlap for the Helmholtz equation. We show that the key ingredients for such an algorithm are the transmission conditions. We derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps. These conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the Schwarz method. This leads to an algorithm in the class of optimized Schwarz methods. We present an asymptotic analysis of the optimized Schwarz method for two types of transmission conditions, Robin conditions and transmission conditions with second order tangential derivatives. Numerical results illustrate the effectiveness of the optimized Schwarz method on a model problem and on a problem from industry.
A nonoverlapping domain decomposition method for Maxwell’s equations in three dimensions
 SIAM J. Numer. Anal
"... Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with muc ..."
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Cited by 35 (10 self)
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Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with much simpler coarse solvers. Though the condition number of the preconditioned system may not have a good bound, we are able to show that the convergence rate of the PCG method with such substructuring preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficient in the elliptic equation. 1.
Nonlinearly preconditioned inexact Newton algorithms
 SIAM J. Sci. Comput
, 2000
"... Abstract. Inexact Newton algorithms are commonlyused for solving large sparse nonlinear system of equations F (u ∗ ) = 0 arising, for example, from the discretization of partial differential equations. Even with global strategies such as linesearch or trust region, the methods often stagnate at loc ..."
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Cited by 35 (14 self)
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Abstract. Inexact Newton algorithms are commonlyused for solving large sparse nonlinear system of equations F (u ∗ ) = 0 arising, for example, from the discretization of partial differential equations. Even with global strategies such as linesearch or trust region, the methods often stagnate at local minima of �F �, especiallyfor problems with unbalanced nonlinearities, because the methods do not have builtin machineryto deal with the unbalanced nonlinearities. To find the same solution u ∗ , one maywant to solve instead an equivalent nonlinearlypreconditioned system F(u ∗ ) = 0 whose nonlinearities are more balanced. In this paper, we propose and studya nonlinear additive Schwarzbased parallel nonlinear preconditioner and show numericallythat the new method converges well even for some difficult problems, such as high Reynolds number flows, where a traditional inexact Newton method fails. Key words. nonlinear preconditioning, inexact Newton methods, Krylov subspace methods, nonlinear additive Schwarz, domain decomposition, nonlinear equations, parallel computing, incompressible