An Introduction to Domain Decomposition Methods

Dolean, Victorita; Jolivet, Pierre; Nataf, F.

doi:10.1137/1.9781611974065

Cited by 259 publications

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“…Domain decomposition methods are particularly well-adapted for the parallel solution of linear systems that appear in finite difference and finite element methods. Among the various domain decomposition methods [24,28], we focus our attention here on the Classical Schwarz Waveform Relaxation (CSWR) DDM [1,13,25,26,27,28,29,30,31,32,36]. Even if this method has received much attention over the past years for many applications, to the best of the authors' knowledge, the first application to the Schrödinger equation can be found in [32].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

2018

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This paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for variable coefficients linear and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [12] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis.

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Section: Introductionmentioning

confidence: 99%

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

2018

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“…The second inequality in (5.9) can be proved, e.g., using the results well-established in the domain decomposition methods; see, e.g., [20,Section 7.8]. A purely algebraic proof is given in [44, Section 5.2].…”

Section: Bound Using the L 2 Norm Of The Residual Representationmentioning

confidence: 99%

Estimating and localizing the algebraic and total numerical errors using flux reconstructions

2017

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This paper presents a methodology for computing upper and lower bounds for both the algebraic and total errors in the context of the conforming finite element discretization of the Poisson model problem and an arbitrary iterative algebraic solver. The derived bounds do not contain any unspecified constants and allow estimating the local distribution of both errors over the computational domain. Combining these bounds, we also obtain guaranteed upper and lower bounds on the discretization error. This allows to propose novel mathematically justified stopping criteria for iterative algebraic solvers ensuring that the algebraic error will lie below the discretization one. Our upper algebraic and total error bounds are based on locally reconstructed fluxes in H(div, Ω), whereas the lower algebraic and total error bounds rely on locally constructed H 1 0 (Ω)-liftings of the algebraic and total residuals. We prove global and local efficiency of the upper bound on the total error and its robustness with respect to the approximation polynomial degree. Relationships to the previously published estimates on the algebraic error are discussed. Theoretical results are illustrated on numerical experiments for higher-order finite element approximations and the preconditioned conjugate gradient method. They in particular witness that the proposed methodology yields a tight estimate on the local distribution of the algebraic and total errors over the computational domain and illustrate the associate cost.Keywords: Numerical solution of partial differential equations, finite element method, a posteriori error estimation, algebraic error, discretization error, stopping criteria, spatial distribution of the error MSC: 65N15, 65N30, 76M10, 65N22, 65F10.

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“…Because we use complex-valued scalars, hypre, and in particular its Maxwell solver AMS [72], cannot be used. MueLu [73] Trilinos can only solve Maxwell's equation in eddy current formulation 6 , and it is not clear how it handles high-order edge elements. Figure 7a is a plot of the time to solution, including both the setup and the solution phases, for solving the linear system on 512 up to 4,096 subdomains.…”

Section: Scaling Analysismentioning

confidence: 99%

“…Variants of (pseudo-)block methods guarantee at least a speedup of nearly 2. We notice that the best approach in terms 6 which assumes that µ 0 ω 2 εE = 0 in eq. (5) Alternative of computation time is 7), which is a combination of recycling and block methods.…”

Section: Scaling Analysismentioning

confidence: 99%

“…In recent years, a lot of efforts have been made to design highly scalable preconditioners for computational fluid dynamics [1], [2] or solid mechanics [3], [4], for example with multigrid [5] or domain decomposition [6] methods. Most of these advanced preconditioners, however, rely on basic iterative methods, such as the Generalized Minimal RESidual method [7] (GMRES) or the Preconditioned Conjugate Gradient [8] (PCG).…”

Section: Introductionmentioning

confidence: 99%

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Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers

Jolivet¹,

Tournier

2016

SC16: International Conference for High Performance Computing, Networking, Storage and Analysis

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An Introduction to Domain Decomposition Methods

Abstract: InfiniBand is a registered trademark of the InfiniBand Trade Association.Intel is a registered trademark of Intel Corporation or its subsidiaries in the United States and other countries.

Cited by 259 publications

References 0 publications

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

Estimating and localizing the algebraic and total numerical errors using flux reconstructions

Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers

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