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Volume 38, Issue 5
Non-Iterative Parallel Schwarz Domain Decomposition Algorithms for a Two-Domain Parabolic Problem

Minjie Shen, Danping Yang & Haibiao Zheng

DOI: 10.4208/cicp.OA-2024-0085

Commun. Comput. Phys., 38 (2025), pp. 1552-1584.

Published online: 2025-09

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  • Abstract

Based on overlapping domain decomposition, two parallel Schwarz finite element algorithms are presented for solving a two-domain parabolic problem. Our algorithms are based on two ideas. One is the observation that the influence of the non-homogeneous boundary data diminishes exponentially towards the interior of the subdomains. Another is to apply proper partitions of unity to distribute the residuals of the discretized systems reasonably into many subdomains so as to avoid repeated correction in the overlapping parts. The resulting algorithms are fully parallel in each subdomain. At each time step, only one iteration step is required to reach the optimal order of accuracy. The convergence rate depending on the mesh parameters are analyzed. Finally, numerical results are reported, which verify the theoretical analysis.

  • Keywords

Domain decomposition, parallel Schwarz method, parabolic problems, error estimate.

  • AMS Subject Headings

65M55, 65M60, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-38-1552, author = {Shen , MinjieYang , Danping and Zheng , Haibiao}, title = {Non-Iterative Parallel Schwarz Domain Decomposition Algorithms for a Two-Domain Parabolic Problem}, journal = {Communications in Computational Physics}, year = {2025}, volume = {38}, number = {5}, pages = {1552--1584}, abstract = {

Based on overlapping domain decomposition, two parallel Schwarz finite element algorithms are presented for solving a two-domain parabolic problem. Our algorithms are based on two ideas. One is the observation that the influence of the non-homogeneous boundary data diminishes exponentially towards the interior of the subdomains. Another is to apply proper partitions of unity to distribute the residuals of the discretized systems reasonably into many subdomains so as to avoid repeated correction in the overlapping parts. The resulting algorithms are fully parallel in each subdomain. At each time step, only one iteration step is required to reach the optimal order of accuracy. The convergence rate depending on the mesh parameters are analyzed. Finally, numerical results are reported, which verify the theoretical analysis.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2024-0085}, url = {http://global-sci.org/intro/article_detail/cicp/24466.html} }
TY - JOUR T1 - Non-Iterative Parallel Schwarz Domain Decomposition Algorithms for a Two-Domain Parabolic Problem AU - Shen , Minjie AU - Yang , Danping AU - Zheng , Haibiao JO - Communications in Computational Physics VL - 5 SP - 1552 EP - 1584 PY - 2025 DA - 2025/09 SN - 38 DO - http://doi.org/10.4208/cicp.OA-2024-0085 UR - https://global-sci.org/intro/article_detail/cicp/24466.html KW - Domain decomposition, parallel Schwarz method, parabolic problems, error estimate. AB -

Based on overlapping domain decomposition, two parallel Schwarz finite element algorithms are presented for solving a two-domain parabolic problem. Our algorithms are based on two ideas. One is the observation that the influence of the non-homogeneous boundary data diminishes exponentially towards the interior of the subdomains. Another is to apply proper partitions of unity to distribute the residuals of the discretized systems reasonably into many subdomains so as to avoid repeated correction in the overlapping parts. The resulting algorithms are fully parallel in each subdomain. At each time step, only one iteration step is required to reach the optimal order of accuracy. The convergence rate depending on the mesh parameters are analyzed. Finally, numerical results are reported, which verify the theoretical analysis.

Shen , MinjieYang , Danping and Zheng , Haibiao. (2025). Non-Iterative Parallel Schwarz Domain Decomposition Algorithms for a Two-Domain Parabolic Problem. Communications in Computational Physics. 38 (5). 1552-1584. doi:10.4208/cicp.OA-2024-0085
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