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Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils

Received: 30 May 2013     Published: 10 August 2013
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Abstract

Consistency of the Douglas – Rachford dimensional splitting scheme is proved for the sum of three nonlinear operators constituting an evolution equation. It is shown that the operators must be densely defined, maximal monotone and single valued on a real Hilbert space in order to satisfy conditions, under which the splitting algorithm can be applied. Numerical experiment conducted for a three-dimensional Stefan problem in permafrost soils suggests that the Douglas – Rachford scheme produces reasonable results, although the convergence rate remains unestablished.

Published in Applied and Computational Mathematics (Volume 2, Issue 4)
DOI 10.11648/j.acm.20130204.11
Page(s) 100-108
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2013. Published by Science Publishing Group

Keywords

Splitting Algorithms, Alternating Directions Scheme, Stefan Problem, Nonlinear Heat Equation, Consistent Approximation

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    Taras A. Dauzhenka, Igor A. Gishkeluk. (2013). Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils. Applied and Computational Mathematics, 2(4), 100-108. https://doi.org/10.11648/j.acm.20130204.11

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    Taras A. Dauzhenka; Igor A. Gishkeluk. Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils. Appl. Comput. Math. 2013, 2(4), 100-108. doi: 10.11648/j.acm.20130204.11

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    AMA Style

    Taras A. Dauzhenka, Igor A. Gishkeluk. Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils. Appl Comput Math. 2013;2(4):100-108. doi: 10.11648/j.acm.20130204.11

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  • @article{10.11648/j.acm.20130204.11,
      author = {Taras A. Dauzhenka and Igor A. Gishkeluk},
      title = {Consistency of the Douglas – Rachford Splitting Algorithm for the Sum of Three Nonlinear Operators: Application to the Stefan Problem in Permafrost Soils},
      journal = {Applied and Computational Mathematics},
      volume = {2},
      number = {4},
      pages = {100-108},
      doi = {10.11648/j.acm.20130204.11},
      url = {https://doi.org/10.11648/j.acm.20130204.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20130204.11},
      abstract = {Consistency of the Douglas – Rachford dimensional splitting scheme is proved for the sum of three nonlinear operators constituting an evolution equation. It is shown that the operators must be densely defined, maximal monotone and single valued on a real Hilbert space in order to satisfy conditions, under which the splitting algorithm can be applied. Numerical experiment conducted for a three-dimensional Stefan problem in permafrost soils suggests that the Douglas – Rachford scheme produces reasonable results, although the convergence rate remains unestablished.},
     year = {2013}
    }
    

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    AB  - Consistency of the Douglas – Rachford dimensional splitting scheme is proved for the sum of three nonlinear operators constituting an evolution equation. It is shown that the operators must be densely defined, maximal monotone and single valued on a real Hilbert space in order to satisfy conditions, under which the splitting algorithm can be applied. Numerical experiment conducted for a three-dimensional Stefan problem in permafrost soils suggests that the Douglas – Rachford scheme produces reasonable results, although the convergence rate remains unestablished.
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