A posteriori error analysis for Schwarz overlapping domain decomposition methods

Chaudhry, Jehanzeb H.; Estep, Donald; Tavener, Simon

doi:10.48550/arxiv.1907.01139

Cited by 6 publications

(12 citation statements)

References 21 publications

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“…The analysis presented here extends earlier work on the a posteriori error analysis of the Parareal algorithm in [7], and incorporates the work of [4] on the a posteriori error analysis of overlapping Schwarz domain decomposition algorithms to provide an estimate of the spatial discretization error. One significant difference between the earlier work addressing the time integration of ODEs using the Parareal method and the current work addressing PDEs, is that it is no longer assumed on the fine time scale, that an implicit time integration method can be solved exactly, but rather, time integration on the fine-scale may require a further iterative solution strategy in space.…”

Section: Introductionmentioning

confidence: 74%

“…2. We estimate E II using the a posteriori error analysis for overlapping domain decomposition presented in [4]. This analysis allows us to further split E II into iterative and discretization components.…”

Section: Overview Of the Strategymentioning

confidence: 99%

“…The analysis of [4] is used to estimate the error term E II . We define the global (spatial) adjoint problem: find φ p n ∈ H 1 0 (Ω) such that,…”

Section: Estimating E IImentioning

confidence: 99%

“…The proof of Theorem 2 is provided in [4]. The term D p s,n quantifies the discretization error contribution (that is, due to using the spaces V qs i,ks ) while the term D p k,n quantifies the domain decomposition iteration error contribution (that is, due to using a finite number K s iterations).…”

Section: Estimating E IImentioning

confidence: 99%

See 3 more Smart Citations

A posteriori error analysis for a space-time parallel discretization of parabolic partial differential equations

Chaudhry¹,

Estep²,

Tavener³

2021

Preprint

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We construct a space-time parallel method for solving parabolic partial differential equations by coupling the Parareal algorithm in time with overlapping domain decomposition in space. Reformulating the original Parareal algorithm as a variational method and implementing a finite element discretization in space enables an adjoint-based a posteriori error analysis to be performed. Through an appropriate choice of adjoint problems and residuals the error analysis distinguishes between errors arising due to the temporal and spatial discretizations, as well as between the errors arising due to incomplete Parareal iterations and incomplete iterations of the domain decomposition solver. We first develop an error analysis for the Parareal method applied to parabolic partial differential equations, and then refine this analysis to the case where the associated spatial problems are solved using overlapping domain decomposition. These constitute our Time Parallel Algorithm (TPA) and Space-Time Parallel Algorithm (STPA) respectively. Numerical experiments demonstrate the accuracy of the estimator for both algorithms and the iterations between distinct components of the error.

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Section: Introductionmentioning

confidence: 74%

Section: Overview Of the Strategymentioning

confidence: 99%

“…The analysis of [4] is used to estimate the error term E II . We define the global (spatial) adjoint problem: find φ p n ∈ H 1 0 (Ω) such that,…”

Section: Estimating E IImentioning

confidence: 99%

Section: Estimating E IImentioning

confidence: 99%

See 2 more Smart Citations

A posteriori error analysis for a space-time parallel discretization of parabolic partial differential equations

Chaudhry¹,

Estep²,

Tavener³

2021

Preprint

Self Cite

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show abstract

“…Classical a posteriori error analysis deals with QoIs that can be expressed as bounded functionals of the solution and has been widely explored [1][2][3][4][6][7][8][9][10][11][12][13]15,16,[19][20][21]24,26,28,30,31,35]. The error estimation utilizes generalized Green's functions solving an adjoint problem, computable residuals of the numerical solution, and variational analysis [1,4,21,27,30,31].…”

Section: Introductionmentioning

confidence: 99%

Error estimation for the time to a threshold value in evolutionary partial differential equations

Chaudhry¹,

Estep²,

Giannini³

et al. 2021

Preprint

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We develop an a posteriori error analysis for a novel quantity of interest (QoI) evolutionary partial differential equations (PDEs). Specifically, the QoI is the first time at which a functional of the solution to the PDE achieves a threshold value signifying a particular event, and differs from classical QoIs which are modeled as bounded linear functionals. We use Taylor's theorem and adjoint based analysis to derive computable and accurate error estimates for linear parabolic and hyperbolic PDEs. Specifically, the heat equation and linearized shallow water equations (SWE) are used for the parabolic and hyperbolic cases, respectively. Numerical examples illustrate the accuracy of the error estimates.

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A posteriori error analysis for a space‐time parallel discretization of parabolic partial differential equations

Chaudhry,

Estep,

Tavener

2023

Numerical Methods Partial

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We construct a space‐time parallel method for solving parabolic partial differential equations by coupling the parareal algorithm in time with overlapping domain decomposition in space. The goal is to obtain a discretization consisting of “local” problems that can be solved on parallel computers efficiently. However, this introduces significant sources of error that must be evaluated. Reformulating the original parareal algorithm as a variational method and implementing a finite element discretization in space enables an adjoint‐based a posteriori error analysis to be performed. Through an appropriate choice of adjoint problems and residuals the error analysis distinguishes between errors arising due to the temporal and spatial discretizations, as well as between the errors arising due to incomplete parareal iterations and incomplete iterations of the domain decomposition solver. We first develop an error analysis for the parareal method applied to parabolic partial differential equations, and then refine this analysis to the case where the associated spatial problems are solved using overlapping domain decomposition. These constitute our time parallel algorithm and space‐time parallel algorithm respectively. Numerical experiments demonstrate the accuracy of the estimator for both algorithms and the iterations between distinct components of the error.

show abstract

A posteriori error analysis for Schwarz overlapping domain decomposition methods

Cited by 6 publications

References 21 publications

A posteriori error analysis for a space-time parallel discretization of parabolic partial differential equations

A posteriori error analysis for a space-time parallel discretization of parabolic partial differential equations

Error estimation for the time to a threshold value in evolutionary partial differential equations

A posteriori error analysis for a space‐time parallel discretization of parabolic partial differential equations

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