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Non-iterative parallel Schwarz algorithms based on overlapping domain decomposition for parabolic partial differential equations

Author: Danping Yang
Journal: Math. Comp. 86 (2017), 2687-2718
MSC (2010): Primary 65N30, 65F10
Published electronically: May 11, 2017
MathSciNet review: 3667021
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Abstract: Two non-iterative parallel Schwarz algorithms (NIPSA) are presented to solve initial-boundary value problems of parabolic partial differential equations of second order. Algorithms are based on an overlapping domain decomposition and are fully parallel. A new idea is to introduce a partition of unity to distribute reasonably residuals of systems into sub-domains in the first algorithm and to sum weighted local corrections of solutions on sub-domains in the second one. Theoretical analysis shows that the algorithms have very good approximate property. At each time step, no iteration is required to reach the optimal order accuracy in $ L^2$-norm. As well small overlapping can be used under some conditions for domain decomposition. Numerical results are also reported, which verify the theoretical analysis.

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Danping Yang
Affiliation: Department of Mathematics, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, and NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, East China Normal University, Shanghai, 200062, People’s Republic of China

Received by editor(s): February 5, 2014
Received by editor(s) in revised form: March 17, 2015, and July 7, 2015
Published electronically: May 11, 2017
Additional Notes: This research was supported partially by the National Natural Science Foundation of China under the grants 11571115 and 11171113 and by the Science and Technology Commission of Shanghai Municipality, grant No. 13dz2260400.
Article copyright: © Copyright 2017 American Mathematical Society