A note on the H1-convergence of the overlapping Schwarz waveform relaxation method for the heat equation

Hui Zhang; Yao Lin Jiang

doi:10.1007/s11075-013-9734-7

A note on the H¹-convergence of the overlapping Schwarz waveform relaxation method for the heat equation

Hui Zhang, Yao Lin Jiang^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

Abstract

The overlapping Schwarz waveform relaxation method is a parallel iterative method for solving time-dependent PDEs. Convergence of the method for the linear heat equation has been studied under infinity norm but it was unknown under the energy norm at the continuous level. The question is interesting for applications concerning fluxes or gradients of the solutions. In this work, we show that the energy norm of the errors of iterates is bounded by their infinity norm. Therefore, we give an affirmative answer to this question for the first time.

Original language	English
Pages (from-to)	299-307
Number of pages	9
Journal	Numerical Algorithms
Volume	66
Issue number	2
DOIs	https://doi.org/10.1007/s11075-013-9734-7
Publication status	Published - Jun 2014
Externally published	Yes

Keywords

Convergence
Energy norm
Schwarz waveform relaxation

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title = "A note on the H1-convergence of the overlapping Schwarz waveform relaxation method for the heat equation",

abstract = "The overlapping Schwarz waveform relaxation method is a parallel iterative method for solving time-dependent PDEs. Convergence of the method for the linear heat equation has been studied under infinity norm but it was unknown under the energy norm at the continuous level. The question is interesting for applications concerning fluxes or gradients of the solutions. In this work, we show that the energy norm of the errors of iterates is bounded by their infinity norm. Therefore, we give an affirmative answer to this question for the first time.",

keywords = "Convergence, Energy norm, Schwarz waveform relaxation",

author = "Hui Zhang and Jiang, {Yao Lin}",

note = "Funding Information: The work was supported by the Natural Science Foundation of China (11071192) and the International Science and Technology Cooperation Program of China (2010DFA14700).",

year = "2014",

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AU - Zhang, Hui

AU - Jiang, Yao Lin

N1 - Funding Information: The work was supported by the Natural Science Foundation of China (11071192) and the International Science and Technology Cooperation Program of China (2010DFA14700).

PY - 2014/6

Y1 - 2014/6

N2 - The overlapping Schwarz waveform relaxation method is a parallel iterative method for solving time-dependent PDEs. Convergence of the method for the linear heat equation has been studied under infinity norm but it was unknown under the energy norm at the continuous level. The question is interesting for applications concerning fluxes or gradients of the solutions. In this work, we show that the energy norm of the errors of iterates is bounded by their infinity norm. Therefore, we give an affirmative answer to this question for the first time.

AB - The overlapping Schwarz waveform relaxation method is a parallel iterative method for solving time-dependent PDEs. Convergence of the method for the linear heat equation has been studied under infinity norm but it was unknown under the energy norm at the continuous level. The question is interesting for applications concerning fluxes or gradients of the solutions. In this work, we show that the energy norm of the errors of iterates is bounded by their infinity norm. Therefore, we give an affirmative answer to this question for the first time.

KW - Convergence

KW - Energy norm

KW - Schwarz waveform relaxation

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JF - Numerical Algorithms

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ER -

A note on the H¹-convergence of the overlapping Schwarz waveform relaxation method for the heat equation

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Keywords

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