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47
Domain Decomposition Algorithms With Small Overlap
, 1994
"... Numerical experiments have shown that twolevel Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear ..."
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Cited by 87 (11 self)
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Numerical experiments have shown that twolevel Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear system of equations that remains after that the variables interior to the subregions have been eliminated. In this paper, a supporting theory is developed.
Overlapping Schwarz waveform relaxation for convectiondominated nonlinear conservation laws
 SIAM J. Sci. Comput
, 2002
"... Abstract. We analyze the convergence of the overlapping Schwarz waveform relaxation algorithm applied to convectiondominated nonlinear conservation laws in one spatial dimension. For two subdomains and bounded time intervals we prove superlinear asymptotic convergence of the algorithm in the parabo ..."
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Cited by 12 (4 self)
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Abstract. We analyze the convergence of the overlapping Schwarz waveform relaxation algorithm applied to convectiondominated nonlinear conservation laws in one spatial dimension. For two subdomains and bounded time intervals we prove superlinear asymptotic convergence of the algorithm in the parabolic case and convergence in a finite number of steps in the hyperbolic limit. The convergence results depend on the overlap, the viscosity, and the length of the time interval under consideration, but they are independent of the number of subdomains, as a generalization of the results to many subdomains shows. To investigate the behavior of the algorithm for a long time, we apply it to the Burgers equation and use a steady state argument to prove that the algorithm converges linearly over long time intervals. This result reveals an interesting paradox: while for the superlinear convergence rate on bounded time intervals decreasing the viscosity improves the performance, in the linear convergence regime decreasing the viscosity slows down the convergence rate and the algorithm can converge arbitrarily slowly, if there is a standing shock wave in the overlap. We illustrate our theoretical results with numerical experiments.
Overlapping Schwarz Waveform Relaxation for Parabolic Problems
 In Proceedings of Algoritmy'97
, 1997
"... Introduction We analyze a new domain decomposition algorithm to solve parabolic partial differential equations. Two classical approaches can be found in the literature: 1. Discretizing time and applying domain decomposition to the obtained sequence of elliptic problems, like in [Kuz90], [Meu91], [ ..."
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Cited by 10 (4 self)
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Introduction We analyze a new domain decomposition algorithm to solve parabolic partial differential equations. Two classical approaches can be found in the literature: 1. Discretizing time and applying domain decomposition to the obtained sequence of elliptic problems, like in [Kuz90], [Meu91], [Cai91] and references therein. 2. Discretizing space and applying waveform relaxation to the large system of ordinary differential equations, like in [LO87], [JP95] and [JV96] and references therein. In contrary to the classical approaches, we formulate a parallel solution algorithm without any discretization. We decompose the domain into overlapping subdomains in space and we consider the parabolic problem on each subdomain over a given time interval. We solve iteratively parabolic problems on subdomains, exchanging boundary information at the interfaces of subdomains. This algorithm is like a classical overlapping additive Schwarz, but on subdomains, a time dependent problem is sol
PRECONDITIONING DISCRETIZATIONS OF SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS
, 2009
"... This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for approach to the construction of preconditioners for unbounded saddle point problems. To illustrate this approach a number of examples will be c ..."
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Cited by 10 (2 self)
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This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for approach to the construction of preconditioners for unbounded saddle point problems. To illustrate this approach a number of examples will be considered. In particular, parameter dependent systems arising in areas like incompressible flow, linear elasticity, and optimal control theory will be studied. The paper contains analysis of several examples and models which have been discussed in the literature previously. However, here each example is discussed with reference to a more unified abstract approach.
Absorbing boundary conditions for the wave equation and parallel
 computing, Math.ofComp.74 (2004), no. 249, 153–176. MR2085406 (2005h:65158
"... Abstract. Absorbing boundary conditions have been developed for various types of problems to truncate infinite domains in order to perform computations. But absorbing boundary conditions have a second, recent and important application: parallel computing. We show that absorbing boundary conditions a ..."
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Cited by 9 (6 self)
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Abstract. Absorbing boundary conditions have been developed for various types of problems to truncate infinite domains in order to perform computations. But absorbing boundary conditions have a second, recent and important application: parallel computing. We show that absorbing boundary conditions are essential for a good performance of the Schwarz waveform relaxation algorithm applied to the wave equation. In turn this application gives the idea of introducing a layer close to the truncation boundary which leads to a new way of optimizing absorbing boundary conditions for truncating domains. We optimize the conditions in the case of straight boundaries and illustrate our analysis with numerical experiments both for truncating domains and the Schwarz waveform relaxation algorithm. 1.
Optimized multiplicative, additive and restricted additive schwarz preconditioning. in preparation
, 2005
"... Abstract. We demonstrate that a small modification of the multiplicative, additive and restricted additive Schwarz preconditioner at the algebraic level, motivated by optimized Schwarz methods defined at the continuous level, leads to a significant reduction in the iteration count of the iterative s ..."
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Cited by 8 (4 self)
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Abstract. We demonstrate that a small modification of the multiplicative, additive and restricted additive Schwarz preconditioner at the algebraic level, motivated by optimized Schwarz methods defined at the continuous level, leads to a significant reduction in the iteration count of the iterative solver. Numerical experiments using finite difference and spectral element discretizations of the positive definite Helmholtz problem and an idealized climate simulation illustrate the effectiveness of the new approach. Key words. Domain decomposition, restricted additive Schwarz method, optimized Schwarz methods, multiplicative Schwarz, spectral elements, highorder. AMS subject classifications. 65F19, 65N22, 65N35 1. Introduction. The
Domain splitting algorithm for mixed finite element approximations to parabolic problems
 EastWest J. Numer. Math
, 1996
"... Abstract  In this paper we fromulate and study a domain decomposition algorithm for solving mixed nite element approximations to parabolic initialboundary value problems. In contrast to the usual overlapping domain decomposition method this technique leads to noniterative algorithms, i.e. the subd ..."
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Cited by 7 (0 self)
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Abstract  In this paper we fromulate and study a domain decomposition algorithm for solving mixed nite element approximations to parabolic initialboundary value problems. In contrast to the usual overlapping domain decomposition method this technique leads to noniterative algorithms, i.e. the subdomain problems are solved independently and the solution in the whole domain is obtained from the local solutions by restriction and simple averaging. The algorithm exploits the fact that the time discretization leads to an elliptic problem with a large positive coe cient infrontof the zero order term. The solutions of such problems exhibit a boundary layer with thickness proportional to the square root of the time discretization parameter. Thus, any error in the boundary conditions will decay exponentially and a reasonable overlap will produce a su ciently accurate method. We prove that the proposed algorithm is stable in L2norm and has the same accuracy as the implicit method.
Stabilized explicitimplicit domain decomposition methods for the numerical solution of parabolic equations
 SIAM J. Sci. Comput., Vol
, 2002
"... Abstract. We report a class of stabilized explicitimplicit domain decomposition (SEIDD) methods for the numerical solution of parabolic equations. Explicitimplicit domain decomposition (EIDD) methods are globally noniterative, nonoverlapping domain decomposition methods, which, when compared with ..."
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Cited by 6 (1 self)
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Abstract. We report a class of stabilized explicitimplicit domain decomposition (SEIDD) methods for the numerical solution of parabolic equations. Explicitimplicit domain decomposition (EIDD) methods are globally noniterative, nonoverlapping domain decomposition methods, which, when compared with Schwarzalgorithmbased parabolic solvers, are computationally and communicationally efficient for each simulation time step but suffer from small time step size restrictions. By adding a stabilization step to EIDD, the SEIDD methods retain the timestepwise efficiency in computation and communication of the EIDD methods but exhibit much better numerical stability. Three SEIDD algorithms are presented in this paper, which are experimentally tested to show excellent stability, computation and communication efficiencies, and high parallel speedup and scalability. Key words. parallel computing, nonoverlapping domain decomposition, parabolic equation, globally noniterative method
A balancing domain decomposition method by constraints for advectiondiffusion problems
 Communications in Applied Mathematics and Computational Science
, 2008
"... Abstract. The balancing domain decomposition methods by constraints are extended to solving nonsymmetric, positive definite linear systems resulting from the finite element discretization of advectiondiffusion equations. A preconditioned GMRES iteration is used to solve a Schur complement system o ..."
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Cited by 6 (0 self)
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Abstract. The balancing domain decomposition methods by constraints are extended to solving nonsymmetric, positive definite linear systems resulting from the finite element discretization of advectiondiffusion equations. A preconditioned GMRES iteration is used to solve a Schur complement system of equations for the subdomain interface variables. In the preconditioning step of each iteration, a partially subassembled finite element problem is solved. A convergence rate estimate for the GMRES iteration is established, under the condition that the diameters of subdomains are small enough. It is independent of the number of subdomains and grows only slowly with the subdomain problem size. Numerical experiments for several twodimensional advectiondiffusion problems illustrate the fast convergence of the proposed algorithm. 1.
Local multiplicative Schwarz algorithms for steady and unsteady convectiondiffusion equations
 EASTWEST J. NUMER. MATH
, 1998
"... In this paper, we develop a new class of overlapping Schwarz type algorithms for solving scalar steady and unsteady convectiondiffusion equations discretized by nite element, or finite difference, methods. The preconditioners consist of two components, namely, the usual additive Schwarz preconditio ..."
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Cited by 5 (0 self)
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In this paper, we develop a new class of overlapping Schwarz type algorithms for solving scalar steady and unsteady convectiondiffusion equations discretized by nite element, or finite difference, methods. The preconditioners consist of two components, namely, the usual additive Schwarz preconditioner and the sum of some second order terms constructed by using products of ordered neighboring subdomain preconditioners. The ordering of the subdomain preconditioners is determined by considering the direction of the flow. For the steady case, we prove that the algorithms are optimal in the sense that the convergence rates are independent of the mesh size, as well as the number of subdomains. For the unsteady case, we show the algorithms are optimal without having a coarse space, as long as the time step and the subdomain size satisfy a certain condition. We show by numerical examples that the new algorithms are less sensitive to the direction of the flow than the classical multiplicative Schwarz algorithms, and converge faster than the additive Schwarz algorithms. Thus, the new algorithms are more suitable for fluid flow applications than the classical additive andmultiplicative Schwarz algorithms.